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1502.04623 | 31 | Figure 12. Generated CIFAR images. The rightmost column shows the nearest training examples to the column beside it.
looking objects without overï¬tting (in other words, without copying from the training set). Nonetheless the images in Fig. 12 demonstrate that DRAW is able to capture much of the shape, colour and composition of real photographs.
# 5. Conclusion
This paper introduced the Deep Recurrent Attentive Writer (DRAW) neural network architecture, and demonstrated its ability to generate highly realistic natural images such as photographs of house numbers, as well as improving on the best known results for binarized MNIST generation. We also established that the two-dimensional differentiable at- tention mechanism embedded in DRAW is beneï¬cial not only to image generation, but also to image classiï¬cation.
Figure 11. Training and validation cost on SVHN. The valida- tion cost is consistently lower because the validation set patches were extracted from the image centre (rather than from random locations, as in the training set). The network was never able to overï¬t on the training data.
# Acknowledgments
Of the many who assisted in creating this paper, we are es- pecially thankful to Koray Kavukcuoglu, Volodymyr Mnih, Jimmy Ba, Yaroslav Bulatov, Greg Wayne, Andrei Rusu and Shakir Mohamed. | 1502.04623#31 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 31,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Figure 12. Generated CIFAR images. The rightmost column shows the nearest training examples to the column beside it.\nlooking objects without overï¬tting (in other words, without copying from the training set). Nonetheless the images in Fig. 12 demonstrate that DRAW is able to capture much of the shape, colour and composition of real photographs.\n# 5. Conclusion\nThis paper introduced the Deep Recurrent Attentive Writer (DRAW) neural network architecture, and demonstrated its ability to generate highly realistic natural images such as photographs of house numbers, as well as improving on the best known results for binarized MNIST generation. We also established that the two-dimensional differentiable at- tention mechanism embedded in DRAW is beneï¬cial not only to image generation, but also to image classiï¬cation.\nFigure 11. Training and validation cost on SVHN. The valida- tion cost is consistently lower because the validation set patches were extracted from the image centre (rather than from random locations, as in the training set). The network was never able to overï¬t on the training data.\n# Acknowledgments\nOf the many who assisted in creating this paper, we are es- pecially thankful to Koray Kavukcuoglu, Volodymyr Mnih, Jimmy Ba, Yaroslav Bulatov, Greg Wayne, Andrei Rusu and Shakir Mohamed.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
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] |
1502.04623 | 32 | 2009). CIFAR-10 is very diverse, and with only 50,000 training examples it is very difï¬cult to generate realisticDRAW: A Recurrent Neural Network For Image Generation
# References
Ba, Jimmy, Mnih, Volodymyr, and Kavukcuoglu, Koray. Multiple object recognition with visual attention. arXiv preprint arXiv:1412.7755, 2014.
Dayan, Peter, Hinton, Geoffrey E, Neal, Radford M, and Zemel, Richard S. The helmholtz machine. Neural com- putation, 7(5):889â904, 1995.
Larochelle, Hugo and Murray, Iain. The neural autoregres- sive distribution estimator. Journal of Machine Learning Research, 15:29â37, 2011.
LeCun, Yann, Bottou, L´eon, Bengio, Yoshua, and Haffner, Patrick. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278â 2324, 1998.
Denil, Misha, Bazzani, Loris, Larochelle, Hugo, and de Freitas, Nando. Learning where to attend with deep architectures for image tracking. Neural computation, 24(8):2151â2184, 2012. | 1502.04623#32 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 32,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "2009). CIFAR-10 is very diverse, and with only 50,000 training examples it is very difï¬cult to generate realisticDRAW: A Recurrent Neural Network For Image Generation\n# References\nBa, Jimmy, Mnih, Volodymyr, and Kavukcuoglu, Koray. Multiple object recognition with visual attention. arXiv preprint arXiv:1412.7755, 2014.\nDayan, Peter, Hinton, Geoffrey E, Neal, Radford M, and Zemel, Richard S. The helmholtz machine. Neural com- putation, 7(5):889â904, 1995.\nLarochelle, Hugo and Murray, Iain. The neural autoregres- sive distribution estimator. Journal of Machine Learning Research, 15:29â37, 2011.\nLeCun, Yann, Bottou, L´eon, Bengio, Yoshua, and Haffner, Patrick. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278â 2324, 1998.\nDenil, Misha, Bazzani, Loris, Larochelle, Hugo, and de Freitas, Nando. Learning where to attend with deep architectures for image tracking. Neural computation, 24(8):2151â2184, 2012.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
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1502.04623 | 33 | Mnih, Andriy and Gregor, Karol. Neural variational infer- ence and learning in belief networks. In Proceedings of the 31st International Conference on Machine Learning, 2014.
Gers, Felix A, Schmidhuber, J¨urgen, and Cummins, Fred. Learning to forget: Continual prediction with lstm. Neu- ral computation, 12(10):2451â2471, 2000.
Mnih, Volodymyr, Heess, Nicolas, Graves, Alex, et al. Re- current models of visual attention. In Advances in Neural Information Processing Systems, pp. 2204â2212, 2014.
Julian, Arnoud, Sacha, Multi-digit number recognition from street view imagery using arXiv preprint deep convolutional neural networks. arXiv:1312.6082, 2013.
Graves, Alex. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.
Murray, Iain and Salakhutdinov, Ruslan. Evaluating prob- abilities under high-dimensional latent variable models. In Advances in neural information processing systems, pp. 1137â1144, 2009. | 1502.04623#33 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 33,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Mnih, Andriy and Gregor, Karol. Neural variational infer- ence and learning in belief networks. In Proceedings of the 31st International Conference on Machine Learning, 2014.\nGers, Felix A, Schmidhuber, J¨urgen, and Cummins, Fred. Learning to forget: Continual prediction with lstm. Neu- ral computation, 12(10):2451â2471, 2000.\nMnih, Volodymyr, Heess, Nicolas, Graves, Alex, et al. Re- current models of visual attention. In Advances in Neural Information Processing Systems, pp. 2204â2212, 2014.\nJulian, Arnoud, Sacha, Multi-digit number recognition from street view imagery using arXiv preprint deep convolutional neural networks. arXiv:1312.6082, 2013.\nGraves, Alex. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.\nMurray, Iain and Salakhutdinov, Ruslan. Evaluating prob- abilities under high-dimensional latent variable models. In Advances in neural information processing systems, pp. 1137â1144, 2009.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
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] |
1502.04623 | 34 | Netzer, Yuval, Wang, Tao, Coates, Adam, Bissacco, Alessandro, Wu, Bo, and Ng, Andrew Y. Reading dig- its in natural images with unsupervised feature learning. 2011.
Graves, Alex, Wayne, Greg, and Danihelka, Ivo. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.
Gregor, Karol, Danihelka, Ivo, Mnih, Andriy, Blundell, Charles, and Wierstra, Daan. Deep autoregressive net- works. In Proceedings of the 31st International Confer- ence on Machine Learning, 2014.
Raiko, Tapani, Li, Yao, Cho, Kyunghyun, and Bengio, Yoshua. Iterative neural autoregressive distribution es- timator nade-k. In Advances in Neural Information Pro- cessing Systems, pp. 325â333. 2014.
Ranzato, MarcâAurelio. On learning where to look. arXiv preprint arXiv:1405.5488, 2014.
Hinton, Geoffrey E and Salakhutdinov, Ruslan R. Reduc- ing the dimensionality of data with neural networks. Sci- ence, 313(5786):504â507, 2006. | 1502.04623#34 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 34,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Netzer, Yuval, Wang, Tao, Coates, Adam, Bissacco, Alessandro, Wu, Bo, and Ng, Andrew Y. Reading dig- its in natural images with unsupervised feature learning. 2011.\nGraves, Alex, Wayne, Greg, and Danihelka, Ivo. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.\nGregor, Karol, Danihelka, Ivo, Mnih, Andriy, Blundell, Charles, and Wierstra, Daan. Deep autoregressive net- works. In Proceedings of the 31st International Confer- ence on Machine Learning, 2014.\nRaiko, Tapani, Li, Yao, Cho, Kyunghyun, and Bengio, Yoshua. Iterative neural autoregressive distribution es- timator nade-k. In Advances in Neural Information Pro- cessing Systems, pp. 325â333. 2014.\nRanzato, MarcâAurelio. On learning where to look. arXiv preprint arXiv:1405.5488, 2014.\nHinton, Geoffrey E and Salakhutdinov, Ruslan R. Reduc- ing the dimensionality of data with neural networks. Sci- ence, 313(5786):504â507, 2006.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
} | [
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] |
1502.04623 | 35 | Hochreiter, Sepp and Schmidhuber, J¨urgen. Long short- term memory. Neural computation, 9(8):1735â1780, 1997.
Kingma, Diederik and Ba, Jimmy. method for stochastic optimization. arXiv:1412.6980, 2014. A arXiv preprint Adam:
Rezende, Danilo J, Mohamed, Shakir, and Wierstra, Daan. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st In- ternational Conference on Machine Learning, pp. 1278â 1286, 2014.
Salakhutdinov, Ruslan and Hinton, Geoffrey E. Deep boltz- mann machines. In International Conference on Artiï¬- cial Intelligence and Statistics, pp. 448â455, 2009.
Kingma, Diederik P and Welling, Max. Auto-encoding In Proceedings of the International variational bayes. Conference on Learning Representations (ICLR), 2014.
Salakhutdinov, Ruslan and Murray, Iain. On the quantita- tive analysis of Deep Belief Networks. In Proceedings of the 25th Annual International Conference on Machine Learning, pp. 872â879. Omnipress, 2008.
Krizhevsky, Alex. Learning multiple layers of features from tiny images. 2009. | 1502.04623#35 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 35,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Hochreiter, Sepp and Schmidhuber, J¨urgen. Long short- term memory. Neural computation, 9(8):1735â1780, 1997.\nKingma, Diederik and Ba, Jimmy. method for stochastic optimization. arXiv:1412.6980, 2014. A arXiv preprint Adam:\nRezende, Danilo J, Mohamed, Shakir, and Wierstra, Daan. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31st In- ternational Conference on Machine Learning, pp. 1278â 1286, 2014.\nSalakhutdinov, Ruslan and Hinton, Geoffrey E. Deep boltz- mann machines. In International Conference on Artiï¬- cial Intelligence and Statistics, pp. 448â455, 2009.\nKingma, Diederik P and Welling, Max. Auto-encoding In Proceedings of the International variational bayes. Conference on Learning Representations (ICLR), 2014.\nSalakhutdinov, Ruslan and Murray, Iain. On the quantita- tive analysis of Deep Belief Networks. In Proceedings of the 25th Annual International Conference on Machine Learning, pp. 872â879. Omnipress, 2008.\nKrizhevsky, Alex. Learning multiple layers of features from tiny images. 2009.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
} | [
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] |
1502.04623 | 36 | Krizhevsky, Alex. Learning multiple layers of features from tiny images. 2009.
Salimans, Tim, Kingma, Diederik P, and Welling, Max. Markov chain monte carlo and variational inference: Bridging the gap. arXiv preprint arXiv:1410.6460, 2014.
Larochelle, Hugo and Hinton, Geoffrey E. Learning to combine foveal glimpses with a third-order boltzmann machine. In Advances in Neural Information Processing Systems, pp. 1243â1251. 2010.
Sermanet, Pierre, Frome, Andrea, and Real, Esteban. At- tention for ï¬ne-grained categorization. arXiv preprint arXiv:1412.7054, 2014.
DRAW: A Recurrent Neural Network For Image Generation
Sutskever, Ilya, Vinyals, Oriol, and Le, Quoc VV. Se- In quence to sequence learning with neural networks. Advances in Neural Information Processing Systems, pp. 3104â3112, 2014.
Tang, Yichuan, Srivastava, Nitish, and Salakhutdinov, Rus- lan. Learning generative models with visual attention. arXiv preprint arXiv:1312.6110, 2013. | 1502.04623#36 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 36,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Krizhevsky, Alex. Learning multiple layers of features from tiny images. 2009.\nSalimans, Tim, Kingma, Diederik P, and Welling, Max. Markov chain monte carlo and variational inference: Bridging the gap. arXiv preprint arXiv:1410.6460, 2014.\nLarochelle, Hugo and Hinton, Geoffrey E. Learning to combine foveal glimpses with a third-order boltzmann machine. In Advances in Neural Information Processing Systems, pp. 1243â1251. 2010.\nSermanet, Pierre, Frome, Andrea, and Real, Esteban. At- tention for ï¬ne-grained categorization. arXiv preprint arXiv:1412.7054, 2014.\nDRAW: A Recurrent Neural Network For Image Generation\nSutskever, Ilya, Vinyals, Oriol, and Le, Quoc VV. Se- In quence to sequence learning with neural networks. Advances in Neural Information Processing Systems, pp. 3104â3112, 2014.\nTang, Yichuan, Srivastava, Nitish, and Salakhutdinov, Rus- lan. Learning generative models with visual attention. arXiv preprint arXiv:1312.6110, 2013.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
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] |
1502.04623 | 37 | Tieleman, Tijmen. Optimizing Neural Networks that Gen- erate Images. PhD thesis, University of Toronto, 2014.
Uria, Benigno, Murray, Iain, and Larochelle, Hugo. A deep In Proceedings of the and tractable density estimator. 31st International Conference on Machine Learning, pp. 467â475, 2014.
Zheng, Yin, Zemel, Richard S, Zhang, Yu-Jin, and Larochelle, Hugo. A neural autoregressive approach to attention-based recognition. International Journal of Computer Vision, pp. 1â13, 2014. | 1502.04623#37 | DRAW: A Recurrent Neural Network For Image Generation | This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural
network architecture for image generation. DRAW networks combine a novel
spatial attention mechanism that mimics the foveation of the human eye, with a
sequential variational auto-encoding framework that allows for the iterative
construction of complex images. The system substantially improves on the state
of the art for generative models on MNIST, and, when trained on the Street View
House Numbers dataset, it generates images that cannot be distinguished from
real data with the naked eye. | http://arxiv.org/pdf/1502.04623 | Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra | cs.CV, cs.LG, cs.NE | null | null | cs.CV | 20150216 | 20150520 | [] | {
"authors": "Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, Daan Wierstra",
"chunk_id": 37,
"doc_id": "1502.04623",
"primary_category": "cs.CV",
"published": 20150216,
"source": "http://arxiv.org/pdf/1502.04623",
"summary": "This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural\nnetwork architecture for image generation. DRAW networks combine a novel\nspatial attention mechanism that mimics the foveation of the human eye, with a\nsequential variational auto-encoding framework that allows for the iterative\nconstruction of complex images. The system substantially improves on the state\nof the art for generative models on MNIST, and, when trained on the Street View\nHouse Numbers dataset, it generates images that cannot be distinguished from\nreal data with the naked eye.",
"text": "Tieleman, Tijmen. Optimizing Neural Networks that Gen- erate Images. PhD thesis, University of Toronto, 2014.\nUria, Benigno, Murray, Iain, and Larochelle, Hugo. A deep In Proceedings of the and tractable density estimator. 31st International Conference on Machine Learning, pp. 467â475, 2014.\nZheng, Yin, Zemel, Richard S, Zhang, Yu-Jin, and Larochelle, Hugo. A neural autoregressive approach to attention-based recognition. International Journal of Computer Vision, pp. 1â13, 2014.",
"title": "DRAW: A Recurrent Neural Network For Image Generation",
"year": 2015
} | [
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] |
1502.03167 | 1 | Sergey Ioffe Google Inc., [email protected]
Christian Szegedy Google Inc., [email protected]
# Abstract
Training Deep Neural Networks is complicated by the fact that the distribution of each layerâs inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it no- toriously hard to train models with saturating nonlineari- ties. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer in- puts. Our method draws its strength from making normal- ization a part of the model architecture and performing the normalization for each training mini-batch. Batch Nor- malization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regu- larizer, in some cases eliminating the need for Dropout. Applied to a state-of-the-art image classiï¬cation model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a signiï¬cant margin. Using an ensemble of batch- normalized networks, we improve upon the best published result on ImageNet classiï¬cation: reaching 4.9% top-5 validation error (and 4.8% test error), exceeding the ac- curacy of human raters. | 1502.03167#1 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 1,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Sergey Ioffe Google Inc., [email protected]\nChristian Szegedy Google Inc., [email protected]\n# Abstract\nTraining Deep Neural Networks is complicated by the fact that the distribution of each layerâs inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it no- toriously hard to train models with saturating nonlineari- ties. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer in- puts. Our method draws its strength from making normal- ization a part of the model architecture and performing the normalization for each training mini-batch. Batch Nor- malization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regu- larizer, in some cases eliminating the need for Dropout. Applied to a state-of-the-art image classiï¬cation model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a signiï¬cant margin. Using an ensemble of batch- normalized networks, we improve upon the best published result on ImageNet classiï¬cation: reaching 4.9% top-5 validation error (and 4.8% test error), exceeding the ac- curacy of human raters.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 2 | Using mini-batches of examples, as opposed to one exam- ple at a time, is helpful in several ways. First, the gradient of the loss over a mini-batch is an estimate of the gradient over the training set, whose quality improves as the batch size increases. Second, computation over a batch can be much more efï¬cient than m computations for individual examples, due to the parallelism afforded by the modern computing platforms.
While stochastic gradient is simple and effective, it requires careful tuning of the model hyper-parameters, speciï¬cally the learning rate used in optimization, as well as the initial values for the model parameters. The train- ing is complicated by the fact that the inputs to each layer are affected by the parameters of all preceding layers â so that small changes to the network parameters amplify as the network becomes deeper.
The change in the distributions of layersâ inputs presents a problem because the layers need to continu- ously adapt to the new distribution. When the input dis- tribution to a learning system changes, it is said to experi- ence covariate shift (Shimodaira, 2000). This is typically handled via domain adaptation (Jiang, 2008). However, the notion of covariate shift can be extended beyond the learning system as a whole, to apply to its parts, such as a sub-network or a layer. Consider a network computing
# 1 Introduction | 1502.03167#2 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 2,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Using mini-batches of examples, as opposed to one exam- ple at a time, is helpful in several ways. First, the gradient of the loss over a mini-batch is an estimate of the gradient over the training set, whose quality improves as the batch size increases. Second, computation over a batch can be much more efï¬cient than m computations for individual examples, due to the parallelism afforded by the modern computing platforms.\nWhile stochastic gradient is simple and effective, it requires careful tuning of the model hyper-parameters, speciï¬cally the learning rate used in optimization, as well as the initial values for the model parameters. The train- ing is complicated by the fact that the inputs to each layer are affected by the parameters of all preceding layers â so that small changes to the network parameters amplify as the network becomes deeper.\nThe change in the distributions of layersâ inputs presents a problem because the layers need to continu- ously adapt to the new distribution. When the input dis- tribution to a learning system changes, it is said to experi- ence covariate shift (Shimodaira, 2000). This is typically handled via domain adaptation (Jiang, 2008). However, the notion of covariate shift can be extended beyond the learning system as a whole, to apply to its parts, such as a sub-network or a layer. Consider a network computing\n# 1 Introduction",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 3 | # 1 Introduction
â = F2(F1(u, Î1), Î2)
Deep learning has dramatically advanced the state of the art in vision, speech, and many other areas. Stochas- tic gradient descent (SGD) has proved to be an effec- tive way of training deep networks, and SGD variants such as momentum (Sutskever et al., 2013) and Adagrad (Duchi et al., 2011) have been used to achieve state of the art performance. SGD optimizes the parameters Î of the network, so as to minimize the loss
where F1 and F2 are arbitrary transformations, and the parameters Î1, Î2 are to be learned so as to minimize the loss â. Learning Î2 can be viewed as if the inputs x = F1(u, Î1) are fed into the sub-network
â = F2(x, Î2).
For example, a gradient descent step
Î = arg min Î 1 N N Xi=1 â(xi, Î) | 1502.03167#3 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 3,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 1 Introduction\nâ = F2(F1(u, Î1), Î2)\nDeep learning has dramatically advanced the state of the art in vision, speech, and many other areas. Stochas- tic gradient descent (SGD) has proved to be an effec- tive way of training deep networks, and SGD variants such as momentum (Sutskever et al., 2013) and Adagrad (Duchi et al., 2011) have been used to achieve state of the art performance. SGD optimizes the parameters Î of the network, so as to minimize the loss\nwhere F1 and F2 are arbitrary transformations, and the parameters Î1, Î2 are to be learned so as to minimize the loss â. Learning Î2 can be viewed as if the inputs x = F1(u, Î1) are fed into the sub-network\nâ = F2(x, Î2).\nFor example, a gradient descent step\nÎ = arg min Î 1 N N Xi=1 â(xi, Î)",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
1502.03167 | 5 | (for batch size m and learning rate α) is exactly equivalent to that for a stand-alone network F2 with input x. There- fore, the input distribution properties that make training more efï¬cient â such as having the same distribution be- tween the training and test data â apply to training the sub-network as well. As such it is advantageous for the distribution of x to remain ï¬xed over time. Then, Î2 does
1
not have to readjust to compensate for the change in the distribution of x. | 1502.03167#5 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 5,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "(for batch size m and learning rate α) is exactly equivalent to that for a stand-alone network F2 with input x. There- fore, the input distribution properties that make training more efï¬cient â such as having the same distribution be- tween the training and test data â apply to training the sub-network as well. As such it is advantageous for the distribution of x to remain ï¬xed over time. Then, Î2 does\n1\nnot have to readjust to compensate for the change in the distribution of x.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 6 | Fixed distribution of inputs to a sub-network would have positive consequences for the layers outside the sub- network, as well. Consider a layer with a sigmoid activa- tion function z = g(W u + b) where u is the layer input, the weight matrix W and bias vector b are the layer pa- rameters to be learned, and g(x) = x | increases, gâ²(x) tends to zero. This means that for all di- mensions of x = W u+b except those with small absolute values, the gradient ï¬owing down to u will vanish and the model will train slowly. However, since x is affected by W, b and the parameters of all the layers below, changes to those parameters during training will likely move many dimensions of x into the saturated regime of the nonlin- earity and slow down the convergence. This effect is ampliï¬ed as the network depth increases. In practice, the saturation problem and the resulting vanishing gradi- ents are usually addressed by using Rectiï¬ed Linear Units (Nair & Hinton, 2010) ReLU (x) = max(x, 0), careful initialization (Bengio & Glorot, 2010; Saxe et al., 2013), and small learning | 1502.03167#6 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 6,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Fixed distribution of inputs to a sub-network would have positive consequences for the layers outside the sub- network, as well. Consider a layer with a sigmoid activa- tion function z = g(W u + b) where u is the layer input, the weight matrix W and bias vector b are the layer pa- rameters to be learned, and g(x) = x | increases, gâ²(x) tends to zero. This means that for all di- mensions of x = W u+b except those with small absolute values, the gradient ï¬owing down to u will vanish and the model will train slowly. However, since x is affected by W, b and the parameters of all the layers below, changes to those parameters during training will likely move many dimensions of x into the saturated regime of the nonlin- earity and slow down the convergence. This effect is ampliï¬ed as the network depth increases. In practice, the saturation problem and the resulting vanishing gradi- ents are usually addressed by using Rectiï¬ed Linear Units (Nair & Hinton, 2010) ReLU (x) = max(x, 0), careful initialization (Bengio & Glorot, 2010; Saxe et al., 2013), and small learning",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 8 | We refer to the change in the distributions of internal nodes of a deep network, in the course of training, as In- ternal Covariate Shift. Eliminating it offers a promise of faster training. We propose a new mechanism, which we call Batch Normalization, that takes a step towards re- ducing internal covariate shift, and in doing so dramati- cally accelerates the training of deep neural nets. It ac- complishes this via a normalization step that ï¬xes the means and variances of layer inputs. Batch Normalization also has a beneï¬cial effect on the gradient ï¬ow through the network, by reducing the dependence of gradients on the scale of the parameters or of their initial values. This allows us to use much higher learning rates with- out the risk of divergence. Furthermore, batch normal- ization regularizes the model and reduces the need for Dropout (Srivastava et al., 2014). Finally, Batch Normal- ization makes it possible to use saturating nonlinearities by preventing the network from getting stuck in the satu- rated modes. | 1502.03167#8 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 8,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "We refer to the change in the distributions of internal nodes of a deep network, in the course of training, as In- ternal Covariate Shift. Eliminating it offers a promise of faster training. We propose a new mechanism, which we call Batch Normalization, that takes a step towards re- ducing internal covariate shift, and in doing so dramati- cally accelerates the training of deep neural nets. It ac- complishes this via a normalization step that ï¬xes the means and variances of layer inputs. Batch Normalization also has a beneï¬cial effect on the gradient ï¬ow through the network, by reducing the dependence of gradients on the scale of the parameters or of their initial values. This allows us to use much higher learning rates with- out the risk of divergence. Furthermore, batch normal- ization regularizes the model and reduces the need for Dropout (Srivastava et al., 2014). Finally, Batch Normal- ization makes it possible to use saturating nonlinearities by preventing the network from getting stuck in the satu- rated modes.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 9 | In Sec. 4.2, we apply Batch Normalization to the best- performing ImageNet classiï¬cation network, and show that we can match its performance using only 7% of the training steps, and can further exceed its accuracy by a substantial margin. Using an ensemble of such networks trained with Batch Normalization, we achieve the top-5 error rate that improves upon the best known results on ImageNet classiï¬cation.
2
# 2 Towards
# Reducing
Internal
# Covariate Shift | 1502.03167#9 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 9,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "In Sec. 4.2, we apply Batch Normalization to the best- performing ImageNet classiï¬cation network, and show that we can match its performance using only 7% of the training steps, and can further exceed its accuracy by a substantial margin. Using an ensemble of such networks trained with Batch Normalization, we achieve the top-5 error rate that improves upon the best known results on ImageNet classiï¬cation.\n2\n# 2 Towards\n# Reducing\nInternal\n# Covariate Shift",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 10 | 2
# 2 Towards
# Reducing
Internal
# Covariate Shift
We deï¬ne Internal Covariate Shift as the change in the distribution of network activations due to the change in network parameters during training. To improve the train- ing, we seek to reduce the internal covariate shift. By ï¬xing the distribution of the layer inputs x as the training progresses, we expect to improve the training speed. It has been long known (LeCun et al., 1998b; Wiesler & Ney, 2011) that the network training converges faster if its in- puts are whitened â i.e., linearly transformed to have zero means and unit variances, and decorrelated. As each layer observes the inputs produced by the layers below, it would be advantageous to achieve the same whitening of the in- puts of each layer. By whitening the inputs to each layer, we would take a step towards achieving the ï¬xed distri- butions of inputs that would remove the ill effects of the internal covariate shift. | 1502.03167#10 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 10,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "2\n# 2 Towards\n# Reducing\nInternal\n# Covariate Shift\nWe deï¬ne Internal Covariate Shift as the change in the distribution of network activations due to the change in network parameters during training. To improve the train- ing, we seek to reduce the internal covariate shift. By ï¬xing the distribution of the layer inputs x as the training progresses, we expect to improve the training speed. It has been long known (LeCun et al., 1998b; Wiesler & Ney, 2011) that the network training converges faster if its in- puts are whitened â i.e., linearly transformed to have zero means and unit variances, and decorrelated. As each layer observes the inputs produced by the layers below, it would be advantageous to achieve the same whitening of the in- puts of each layer. By whitening the inputs to each layer, we would take a step towards achieving the ï¬xed distri- butions of inputs that would remove the ill effects of the internal covariate shift.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 11 | We could consider whitening activations at every train- ing step or at some interval, either by modifying the network directly or by changing the parameters of the optimization algorithm to depend on the network ac- tivation values (Wiesler et al., 2014; Raiko et al., 2012; Povey et al., 2014; Desjardins & Kavukcuoglu). How- ever, if these modiï¬cations are interspersed with the op- timization steps, then the gradient descent step may at- tempt to update the parameters in a way that requires the normalization to be updated, which reduces the ef- fect of the gradient step. For example, consider a layer with the input u that adds the learned bias b, and normal- izes the result by subtracting the mean of the activation computed over the training data: E[x] where â is the set of values of x over x = u + b, = the training set, and E[x] = 1 If a gradient N descent step ignores the dependence of E[x] on b, then it P b + âb, where âb will update b x. Then â â E[u + b]. E[u + (b + âb)] = u + b u + (b + âb) b | 1502.03167#11 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 11,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "We could consider whitening activations at every train- ing step or at some interval, either by modifying the network directly or by changing the parameters of the optimization algorithm to depend on the network ac- tivation values (Wiesler et al., 2014; Raiko et al., 2012; Povey et al., 2014; Desjardins & Kavukcuoglu). How- ever, if these modiï¬cations are interspersed with the op- timization steps, then the gradient descent step may at- tempt to update the parameters in a way that requires the normalization to be updated, which reduces the ef- fect of the gradient step. For example, consider a layer with the input u that adds the learned bias b, and normal- izes the result by subtracting the mean of the activation computed over the training data: E[x] where â is the set of values of x over x = u + b, = the training set, and E[x] = 1 If a gradient N descent step ignores the dependence of E[x] on b, then it P b + âb, where âb will update b x. Then â â E[u + b]. E[u + (b + âb)] = u + b u + (b + âb) b",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 12 | will update b x. Then â â E[u + b]. E[u + (b + âb)] = u + b u + (b + âb) b Thus, the combination of the update to b and subsequent change in normalization led to no change in the output of the layer nor, consequently, the loss. As the training continues, b will grow indeï¬nitely while the loss remains ï¬xed. This problem can get worse if the normalization not only centers but also scales the activations. We have ob- served this empirically in initial experiments, where the model blows up when the normalization parameters are computed outside the gradient descent step. | 1502.03167#12 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 12,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "will update b x. Then â â E[u + b]. E[u + (b + âb)] = u + b u + (b + âb) b Thus, the combination of the update to b and subsequent change in normalization led to no change in the output of the layer nor, consequently, the loss. As the training continues, b will grow indeï¬nitely while the loss remains ï¬xed. This problem can get worse if the normalization not only centers but also scales the activations. We have ob- served this empirically in initial experiments, where the model blows up when the normalization parameters are computed outside the gradient descent step.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 13 | The issue with the above approach is that the gradient descent optimization does not take into account the fact that the normalization takes place. To address this issue, we would like to ensure that, for any parameter values, the network always produces activations with the desired distribution. Doing so would allow the gradient of the loss with respect to the model parameters to account for the normalization, and for its dependence on the model parameters Î. Let again x be a layer input, treated as a
vector, and be the set of these inputs over the training data set. The normalization can then be written as a trans- formation
x = Norm(x, )
# X
which depends not only on the given training example x but on all examples â each of which depends on Î if x is generated by another layer. For backpropagation, we would need to compute the Jacobians
âNorm(x, X âx ) and âNorm(x, â X ) ; | 1502.03167#13 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 13,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "The issue with the above approach is that the gradient descent optimization does not take into account the fact that the normalization takes place. To address this issue, we would like to ensure that, for any parameter values, the network always produces activations with the desired distribution. Doing so would allow the gradient of the loss with respect to the model parameters to account for the normalization, and for its dependence on the model parameters Î. Let again x be a layer input, treated as a\nvector, and be the set of these inputs over the training data set. The normalization can then be written as a trans- formation\nx = Norm(x, )\n# X\nwhich depends not only on the given training example x but on all examples â each of which depends on Î if x is generated by another layer. For backpropagation, we would need to compute the Jacobians\nâNorm(x, X âx ) and âNorm(x, â X ) ;",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 14 | # X
ignoring the latter term would lead to the explosion de- scribed above. Within this framework, whitening the layer inputs is expensive, as it requires computing the covari- ance matrix Cov[x] = ExâX [xxT ] E[x]E[x]T and its inverse square root, to produce the whitened activations Cov[x]â1/2(x E[x]), as well as the derivatives of these transforms for backpropagation. This motivates us to seek an alternative that performs input normalization in a way that is differentiable and does not require the analysis of the entire training set after every parameter update.
(e.g. previous (Lyu & Simoncelli, 2008)) use computed over a single training example, or, in the case of image networks, over different feature maps at a given location. However, this changes the representation ability of a network by discarding the absolute scale of activations. We want to a preserve the information in the network, by normalizing the activations in a training example relative to the statistics of the entire training data.
# 3 Normalization via Mini-Batch Statistics | 1502.03167#14 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 14,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# X\nignoring the latter term would lead to the explosion de- scribed above. Within this framework, whitening the layer inputs is expensive, as it requires computing the covari- ance matrix Cov[x] = ExâX [xxT ] E[x]E[x]T and its inverse square root, to produce the whitened activations Cov[x]â1/2(x E[x]), as well as the derivatives of these transforms for backpropagation. This motivates us to seek an alternative that performs input normalization in a way that is differentiable and does not require the analysis of the entire training set after every parameter update.\n(e.g. previous (Lyu & Simoncelli, 2008)) use computed over a single training example, or, in the case of image networks, over different feature maps at a given location. However, this changes the representation ability of a network by discarding the absolute scale of activations. We want to a preserve the information in the network, by normalizing the activations in a training example relative to the statistics of the entire training data.\n# 3 Normalization via Mini-Batch Statistics",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 15 | # 3 Normalization via Mini-Batch Statistics
Since the full whitening of each layerâs inputs is costly and not everywhere differentiable, we make two neces- sary simpliï¬cations. The ï¬rst is that instead of whitening the features in layer inputs and outputs jointly, we will normalize each scalar feature independently, by making it have the mean of zero and the variance of 1. For a layer with d-dimensional input x = (x(1) . . . x(d)), we will nor- malize each dimension
x(k) = x(k) E[x(k)] â Var[x(k)] | 1502.03167#15 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 15,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 3 Normalization via Mini-Batch Statistics\nSince the full whitening of each layerâs inputs is costly and not everywhere differentiable, we make two neces- sary simpliï¬cations. The ï¬rst is that instead of whitening the features in layer inputs and outputs jointly, we will normalize each scalar feature independently, by making it have the mean of zero and the variance of 1. For a layer with d-dimensional input x = (x(1) . . . x(d)), we will nor- malize each dimension\nx(k) = x(k) E[x(k)] â Var[x(k)]",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 16 | p where the expectation and variance are computed over the training data set. As shown in (LeCun et al., 1998b), such normalization speeds up convergence, even when the fea- tures are not decorrelated.
Note that simply normalizing each input of a layer may change what the layer can represent. For instance, nor- malizing the inputs of a sigmoid would constrain them to the linear regime of the nonlinearity. To address this, we make sure that the transformation inserted in the network can represent the identity transform. To accomplish this,
we introduce, for each activation x(k), a pair of parameters γ(k), β(k), which scale and shift the normalized value:
y(k) = γ(k) x(k) + β(k).
These parameters are learned along with the original b model parameters, and restore the representation power of the network. Indeed, by setting γ(k) = Var[x(k)] and β(k) = E[x(k)], we could recover the original activations, if that were the optimal thing to do. | 1502.03167#16 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 16,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "p where the expectation and variance are computed over the training data set. As shown in (LeCun et al., 1998b), such normalization speeds up convergence, even when the fea- tures are not decorrelated.\nNote that simply normalizing each input of a layer may change what the layer can represent. For instance, nor- malizing the inputs of a sigmoid would constrain them to the linear regime of the nonlinearity. To address this, we make sure that the transformation inserted in the network can represent the identity transform. To accomplish this,\nwe introduce, for each activation x(k), a pair of parameters γ(k), β(k), which scale and shift the normalized value:\ny(k) = γ(k) x(k) + β(k).\nThese parameters are learned along with the original b model parameters, and restore the representation power of the network. Indeed, by setting γ(k) = Var[x(k)] and β(k) = E[x(k)], we could recover the original activations, if that were the optimal thing to do.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 17 | In the batch setting where each training step is based on the entire training set, we would use the whole set to nor- malize activations. However, this is impractical when us- ing stochastic optimization. Therefore, we make the sec- ond simpliï¬cation: since we use mini-batches in stochas- tic gradient training, each mini-batch produces estimates of the mean and variance of each activation. This way, the statistics used for normalization can fully participate in the gradient backpropagation. Note that the use of mini- batches is enabled by computation of per-dimension vari- ances rather than joint covariances; in the joint case, reg- ularization would be required since the mini-batch size is likely to be smaller than the number of activations being whitened, resulting in singular covariance matrices.
of size m. Since the normal- ization is applied to each activation independently, let us focus on a particular activation x(k) and omit k for clarity. We have m values of this activation in the mini-batch,
= .
# x1...m} x1...m, and their linear trans{ Let the normalized values be formations be y1...m. We refer to the transform
# B
# b
BNγ,β : x1...m â
y1...m | 1502.03167#17 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 17,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "In the batch setting where each training step is based on the entire training set, we would use the whole set to nor- malize activations. However, this is impractical when us- ing stochastic optimization. Therefore, we make the sec- ond simpliï¬cation: since we use mini-batches in stochas- tic gradient training, each mini-batch produces estimates of the mean and variance of each activation. This way, the statistics used for normalization can fully participate in the gradient backpropagation. Note that the use of mini- batches is enabled by computation of per-dimension vari- ances rather than joint covariances; in the joint case, reg- ularization would be required since the mini-batch size is likely to be smaller than the number of activations being whitened, resulting in singular covariance matrices.\nof size m. Since the normal- ization is applied to each activation independently, let us focus on a particular activation x(k) and omit k for clarity. We have m values of this activation in the mini-batch,\n= .\n# x1...m} x1...m, and their linear trans{ Let the normalized values be formations be y1...m. We refer to the transform\n# B\n# b\nBNγ,β : x1...m â\ny1...m",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 18 | # B
# b
BNγ,β : x1...m â
y1...m
as the Batch Normalizing Transform. We present the BN Transform in Algorithm 1. In the algorithm, Ç« is a constant added to the mini-batch variance for numerical stability.
Input: Values of x over a mini-batch: Parameters to be learned: γ, β ; = x1...m} { B Output: yi = BNγ,β(xi) } { m 1 m // mini-batch mean xi µB â Xi=1 m 1 m Xi=1 xi â Ï2 p xi + β γ Ï2 B µB)2 // mini-batch variance (xi â µB B + Ç« â // normalize xi â b yi â // scale and shift BNγ,β(xi) â¡
# b | 1502.03167#18 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 18,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# B\n# b\nBNγ,β : x1...m â\ny1...m\nas the Batch Normalizing Transform. We present the BN Transform in Algorithm 1. In the algorithm, Ç« is a constant added to the mini-batch variance for numerical stability.\nInput: Values of x over a mini-batch: Parameters to be learned: γ, β ; = x1...m} { B Output: yi = BNγ,β(xi) } { m 1 m // mini-batch mean xi µB â Xi=1 m 1 m Xi=1 xi â Ï2 p xi + β γ Ï2 B µB)2 // mini-batch variance (xi â µB B + Ç« â // normalize xi â b yi â // scale and shift BNγ,β(xi) â¡\n# b",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 20 | 3
indicate that the parameters γ and β are to be learned, but it should be noted that the BN transform does not independently process the activation in each training ex- ample. Rather, BNγ,β(x) depends both on the training example and the other examples in the mini-batch. The scaled and shifted values y are passed to other network layers. The normalized activations x are internal to our transformation, but their presence is crucial. The distri- butions of values of any x has the expected value of 0 and the variance of 1, as long as the elements of each mini-batch are sampled from the same distribution, and if we neglect Ç«. This can be seen by observing that x2 i = 1, and taking expec- x(k) can be viewed as tations. Each normalized activation P b an input to a sub-network composed of the linear trans- b form y(k) = γ(k) x(k) + β(k), followed by the other pro- cessing done by the original network. These sub-network inputs all have ï¬xed means and variances, and although x(k) can change the joint distribution of these normalized over the course of training, we expect that the introduc- tion of normalized inputs accelerates the training of the sub-network and, consequently, the network as a whole. | 1502.03167#20 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 20,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "3\nindicate that the parameters γ and β are to be learned, but it should be noted that the BN transform does not independently process the activation in each training ex- ample. Rather, BNγ,β(x) depends both on the training example and the other examples in the mini-batch. The scaled and shifted values y are passed to other network layers. The normalized activations x are internal to our transformation, but their presence is crucial. The distri- butions of values of any x has the expected value of 0 and the variance of 1, as long as the elements of each mini-batch are sampled from the same distribution, and if we neglect Ç«. This can be seen by observing that x2 i = 1, and taking expec- x(k) can be viewed as tations. Each normalized activation P b an input to a sub-network composed of the linear trans- b form y(k) = γ(k) x(k) + β(k), followed by the other pro- cessing done by the original network. These sub-network inputs all have ï¬xed means and variances, and although x(k) can change the joint distribution of these normalized over the course of training, we expect that the introduc- tion of normalized inputs accelerates the training of the sub-network and, consequently, the network as a whole.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 21 | During training we need to backpropagate the gradi- ent of loss â through this transformation, as well as com- pute the gradients with respect to the parameters of the BN transform. We use chain rule, as follows (before sim- pliï¬cation):
ae _ ae. ae: â By 7 ak _ vm ae . =1/,2 â3/2 Bom = Lint Bayâ (ti â Ms) (OB + â¬)*/ oe _ ym of 1 4 06, hy â2(@:i-HB) due i=1 Oa; ae daz ⢠Ok _ dol 1 (aie Hs + ar = Darâ Torre f + one =v: a ym oo i=1 Oy: op | 1502.03167#21 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 21,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "During training we need to backpropagate the gradi- ent of loss â through this transformation, as well as com- pute the gradients with respect to the parameters of the BN transform. We use chain rule, as follows (before sim- pliï¬cation):\nae _ ae. ae: â By 7 ak _ vm ae . =1/,2 â3/2 Bom = Lint Bayâ (ti â Ms) (OB + â¬)*/ oe _ ym of 1 4 06, hy â2(@:i-HB) due i=1 Oa; ae daz ⢠Ok _ dol 1 (aie Hs + ar = Darâ Torre f + one =v: a ym oo i=1 Oy: op",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 22 | # P
Thus, BN transform is a differentiable transformation that introduces normalized activations into the network. This ensures that as the model is training, layers can continue learning on input distributions that exhibit less internal co- variate shift, thus accelerating the training. Furthermore, the learned afï¬ne transform applied to these normalized activations allows the BN transform to represent the iden- tity transformation and preserves the network capacity.
# 3.1 Training and Inference with Batch- Normalized Networks
To Batch-Normalize a network, we specify a subset of ac- tivations and insert the BN transform for each of them, according to Alg. 1. Any layer that previously received x as the input, now receives BN(x). A model employing Batch Normalization can be trained using batch gradient descent, or Stochastic Gradient Descent with a mini-batch size m > 1, or with any of its variants such as Adagrad
(Duchi et al., 2011). The normalization of activations that depends on the mini-batch allows efï¬cient training, but is neither necessary nor desirable during inference; we want the output to depend only on the input, deterministically. For this, once the network has been trained, we use the normalization
x = E[x] x Var[x] + Ç« â | 1502.03167#22 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 22,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# P\nThus, BN transform is a differentiable transformation that introduces normalized activations into the network. This ensures that as the model is training, layers can continue learning on input distributions that exhibit less internal co- variate shift, thus accelerating the training. Furthermore, the learned afï¬ne transform applied to these normalized activations allows the BN transform to represent the iden- tity transformation and preserves the network capacity.\n# 3.1 Training and Inference with Batch- Normalized Networks\nTo Batch-Normalize a network, we specify a subset of ac- tivations and insert the BN transform for each of them, according to Alg. 1. Any layer that previously received x as the input, now receives BN(x). A model employing Batch Normalization can be trained using batch gradient descent, or Stochastic Gradient Descent with a mini-batch size m > 1, or with any of its variants such as Adagrad\n(Duchi et al., 2011). The normalization of activations that depends on the mini-batch allows efï¬cient training, but is neither necessary nor desirable during inference; we want the output to depend only on the input, deterministically. For this, once the network has been trained, we use the normalization\nx = E[x] x Var[x] + Ç« â",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 23 | # p
# b
using the population, rather than mini-batch, statistics. Neglecting Ç«, these normalized activations have the same mean 0 and variance 1 as during training. We use the un- biased variance estimate Var[x] = m B], where the expectation is over training mini-batches of size m and Ï2 B are their sample variances. Using moving averages in- stead, we can track the accuracy of a model as it trains. Since the means and variances are ï¬xed during inference, the normalization is simply a linear transform applied to each activation. It may further be composed with the scal- ing by γ and shift by β, to yield a single linear transform that replaces BN(x). Algorithm 2 summarizes the proce- dure for training batch-normalized networks. | 1502.03167#23 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 23,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# p\n# b\nusing the population, rather than mini-batch, statistics. Neglecting Ç«, these normalized activations have the same mean 0 and variance 1 as during training. We use the un- biased variance estimate Var[x] = m B], where the expectation is over training mini-batches of size m and Ï2 B are their sample variances. Using moving averages in- stead, we can track the accuracy of a model as it trains. Since the means and variances are ï¬xed during inference, the normalization is simply a linear transform applied to each activation. It may further be composed with the scal- ing by γ and shift by β, to yield a single linear transform that replaces BN(x). Algorithm 2 summarizes the proce- dure for training batch-normalized networks.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
1502.03167 | 24 | Input: Network N with trainable parameters O; subset of activations {a} Output: Batch-normalized network for inference, N24, 1: Ngx <â N_ // Training BN network 2: fork =1...K do 3: Add transformation yâ) = BN) gc) (x 0K)) to sn (Alg. 1) 4: Modify each layer in Nf with input xâ) to take y) instead 5: end for 6: Train Ngy to optimize the parameters © U (9), 80}, 7: Net < Ngy_ // Inference BN network with frozen // parameters 8: fork =1...K do 9: // For clarity, 2 = 2), yÂ¥ =, we = nw, etc. 10: Process multiple training mini-batches B, each of size m, and average over them: E[z] â Es[us] Var[x] â 4 Eg(o3] ll: In N3X, replace the transform y = BN,,g(a) with = ~L.-r+(B- Ele] Â¥ a/Var[x]+e 7 ( at) 12: end for | 1502.03167#24 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 24,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Input: Network N with trainable parameters O; subset of activations {a} Output: Batch-normalized network for inference, N24, 1: Ngx <â N_ // Training BN network 2: fork =1...K do 3: Add transformation yâ) = BN) gc) (x 0K)) to sn (Alg. 1) 4: Modify each layer in Nf with input xâ) to take y) instead 5: end for 6: Train Ngy to optimize the parameters © U (9), 80}, 7: Net < Ngy_ // Inference BN network with frozen // parameters 8: fork =1...K do 9: // For clarity, 2 = 2), yÂ¥ =, we = nw, etc. 10: Process multiple training mini-batches B, each of size m, and average over them: E[z] â Es[us] Var[x] â 4 Eg(o3] ll: In N3X, replace the transform y = BN,,g(a) with = ~L.-r+(B- Ele] Â¥ a/Var[x]+e 7 ( at) 12: end for",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 25 | # 12: end for
Algorithm 2: Training a Batch-Normalized Network
# 3.2 Batch-Normalized Convolutional Net- works
Batch Normalization can be applied to any set of acti- vations in the network. Here, we focus on transforms
4
that consist of an afï¬ne transformation followed by an element-wise nonlinearity:
z = g(W u + b)
where W and b are learned parameters of the model, and ) is the nonlinearity such as sigmoid or ReLU. This for- g( · mulation covers both fully-connected and convolutional layers. We add the BN transform immediately before the nonlinearity, by normalizing x = W u + b. We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity, the shape of its distri- bution is likely to change during training, and constraining its ï¬rst and second moments would not eliminate the co- variate shift. In contrast, W u + b is more likely to have a symmetric, non-sparse distribution, that is âmore Gaus- sianâ (Hyv¨arinen & Oja, 2000); normalizing it is likely to produce activations with a stable distribution. | 1502.03167#25 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 25,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 12: end for\nAlgorithm 2: Training a Batch-Normalized Network\n# 3.2 Batch-Normalized Convolutional Net- works\nBatch Normalization can be applied to any set of acti- vations in the network. Here, we focus on transforms\n4\nthat consist of an afï¬ne transformation followed by an element-wise nonlinearity:\nz = g(W u + b)\nwhere W and b are learned parameters of the model, and ) is the nonlinearity such as sigmoid or ReLU. This for- g( · mulation covers both fully-connected and convolutional layers. We add the BN transform immediately before the nonlinearity, by normalizing x = W u + b. We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity, the shape of its distri- bution is likely to change during training, and constraining its ï¬rst and second moments would not eliminate the co- variate shift. In contrast, W u + b is more likely to have a symmetric, non-sparse distribution, that is âmore Gaus- sianâ (Hyv¨arinen & Oja, 2000); normalizing it is likely to produce activations with a stable distribution.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 26 | Note that, since we normalize W u+b, the bias b can be ignored since its effect will be canceled by the subsequent mean subtraction (the role of the bias is subsumed by β in Alg. 1). Thus, z = g(W u + b) is replaced with
z = g(BN(W u))
where the BN transform is applied independently to each dimension of x = W u, with a separate pair of learned parameters γ(k), β(k) per dimension.
For convolutional layers, we additionally want the nor- malization to obey the convolutional property â so that different elements of the same feature map, at different locations, are normalized in the same way. To achieve this, we jointly normalize all the activations in a mini- be the set of batch, over all locations. In Alg. 1, we let all values in a feature map across both the elements of a mini-batch and spatial locations â so for a mini-batch of q, we use the effec- size m and feature maps of size p tive mini-batch of size mâ² = p q. We learn a pair of parameters γ(k) and β(k) per feature map, rather than per activation. Alg. 2 is modiï¬ed similarly, so that during inference the BN transform applies the same linear transformation to each activation in a given feature map. | 1502.03167#26 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 26,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Note that, since we normalize W u+b, the bias b can be ignored since its effect will be canceled by the subsequent mean subtraction (the role of the bias is subsumed by β in Alg. 1). Thus, z = g(W u + b) is replaced with\nz = g(BN(W u))\nwhere the BN transform is applied independently to each dimension of x = W u, with a separate pair of learned parameters γ(k), β(k) per dimension.\nFor convolutional layers, we additionally want the nor- malization to obey the convolutional property â so that different elements of the same feature map, at different locations, are normalized in the same way. To achieve this, we jointly normalize all the activations in a mini- be the set of batch, over all locations. In Alg. 1, we let all values in a feature map across both the elements of a mini-batch and spatial locations â so for a mini-batch of q, we use the effec- size m and feature maps of size p tive mini-batch of size mâ² = p q. We learn a pair of parameters γ(k) and β(k) per feature map, rather than per activation. Alg. 2 is modiï¬ed similarly, so that during inference the BN transform applies the same linear transformation to each activation in a given feature map.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 27 | # 3.3 Batch Normalization enables higher learning rates
In traditional deep networks, too-high learning rate may result in the gradients that explode or vanish, as well as getting stuck in poor local minima. Batch Normaliza- tion helps address these issues. By normalizing activa- tions throughout the network, it prevents small changes to the parameters from amplifying into larger and subop- timal changes in activations in gradients; for instance, it prevents the training from getting stuck in the saturated regimes of nonlinearities.
Batch Normalization also makes training more resilient to the parameter scale. Normally, large learning rates may increase the scale of layer parameters, which then amplify
5
the gradient during backpropagation and lead to the model explosion. However, with Batch Normalization, back- propagation through a layer is unaffected by the scale of its parameters. Indeed, for a scalar a,
BN(W u) = BN((aW )u)
and we can show that
âBN((aW )u) âu âBN((aW )u) = âBN(W u) âu âBN(W u) âW â(aW ) = 1 a · | 1502.03167#27 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 27,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 3.3 Batch Normalization enables higher learning rates\nIn traditional deep networks, too-high learning rate may result in the gradients that explode or vanish, as well as getting stuck in poor local minima. Batch Normaliza- tion helps address these issues. By normalizing activa- tions throughout the network, it prevents small changes to the parameters from amplifying into larger and subop- timal changes in activations in gradients; for instance, it prevents the training from getting stuck in the saturated regimes of nonlinearities.\nBatch Normalization also makes training more resilient to the parameter scale. Normally, large learning rates may increase the scale of layer parameters, which then amplify\n5\nthe gradient during backpropagation and lead to the model explosion. However, with Batch Normalization, back- propagation through a layer is unaffected by the scale of its parameters. Indeed, for a scalar a,\nBN(W u) = BN((aW )u)\nand we can show that\nâBN((aW )u) âu âBN((aW )u) = âBN(W u) âu âBN(W u) âW â(aW ) = 1 a ·",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 28 | The scale does not affect the layer Jacobian nor, con- sequently, the gradient propagation. Moreover, larger weights lead to smaller gradients, and Batch Normaliza- tion will stabilize the parameter growth.
We further conjecture that Batch Normalization may lead the layer Jacobians to have singular values close to 1, which is known to be beneï¬cial for training (Saxe et al., 2013). Consider two consecutive layers with normalized inputs, and the transformation between these normalized vectors: z are Gaussian x and x is a linear transfor- and uncorrelated, and that F ( b b mation for the given model parameters, then both x and z b x]J T = z] = JCov[ have unit covariances, and I = Cov[ b b JJ T . Thus, JJ T = I, and so all singular values of J b are equal to 1, which preserves the gradient magnitudes during backpropagation. In reality, the transformation is not linear, and the normalized values are not guaranteed to be Gaussian nor independent, but we nevertheless expect Batch Normalization to help make gradient propagation better behaved. The precise effect of Batch Normaliza- tion on gradient propagation remains an area of further study.
# 3.4 Batch Normalization regularizes the model | 1502.03167#28 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 28,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "The scale does not affect the layer Jacobian nor, con- sequently, the gradient propagation. Moreover, larger weights lead to smaller gradients, and Batch Normaliza- tion will stabilize the parameter growth.\nWe further conjecture that Batch Normalization may lead the layer Jacobians to have singular values close to 1, which is known to be beneï¬cial for training (Saxe et al., 2013). Consider two consecutive layers with normalized inputs, and the transformation between these normalized vectors: z are Gaussian x and x is a linear transfor- and uncorrelated, and that F ( b b mation for the given model parameters, then both x and z b x]J T = z] = JCov[ have unit covariances, and I = Cov[ b b JJ T . Thus, JJ T = I, and so all singular values of J b are equal to 1, which preserves the gradient magnitudes during backpropagation. In reality, the transformation is not linear, and the normalized values are not guaranteed to be Gaussian nor independent, but we nevertheless expect Batch Normalization to help make gradient propagation better behaved. The precise effect of Batch Normaliza- tion on gradient propagation remains an area of further study.\n# 3.4 Batch Normalization regularizes the model",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 29 | # 3.4 Batch Normalization regularizes the model
When training with Batch Normalization, a training ex- ample is seen in conjunction with other examples in the mini-batch, and the training network no longer produc- ing deterministic values for a given training example. In our experiments, we found this effect to be advantageous to the generalization of the network. Whereas Dropout (Srivastava et al., 2014) is typically used to reduce over- ï¬tting, in a batch-normalized network we found that it can be either removed or reduced in strength.
# 4 Experiments
# 4.1 Activations over time
To verify the effects of internal covariate shift on train- ing, and the ability of Batch Normalization to combat it, we considered the problem of predicting the digit class on the MNIST dataset (LeCun et al., 1998a). We used a very simple network, with a 28x28 binary image as input, and
1 2 2 0.9 0.8 Without BN With BN 0 0 0.7 10K 20K 30K 40K 50K â2 â2 (a) (b) Without BN (c) With BN | 1502.03167#29 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 29,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 3.4 Batch Normalization regularizes the model\nWhen training with Batch Normalization, a training ex- ample is seen in conjunction with other examples in the mini-batch, and the training network no longer produc- ing deterministic values for a given training example. In our experiments, we found this effect to be advantageous to the generalization of the network. Whereas Dropout (Srivastava et al., 2014) is typically used to reduce over- ï¬tting, in a batch-normalized network we found that it can be either removed or reduced in strength.\n# 4 Experiments\n# 4.1 Activations over time\nTo verify the effects of internal covariate shift on train- ing, and the ability of Batch Normalization to combat it, we considered the problem of predicting the digit class on the MNIST dataset (LeCun et al., 1998a). We used a very simple network, with a 28x28 binary image as input, and\n1 2 2 0.9 0.8 Without BN With BN 0 0 0.7 10K 20K 30K 40K 50K â2 â2 (a) (b) Without BN (c) With BN",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 30 | Figure 1: (a) The test accuracy of the MNIST network the trained with and without Batch Normalization, vs. number of training steps. Batch Normalization helps the network train faster and achieve higher accuracy. (b, c) The evolution of input distributions to a typical sig- moid, over the course of training, shown as th percentiles. Batch Normalization makes the distribution more stable and reduces the internal covariate shift.
3 fully-connected hidden layers with 100 activations each. Each hidden layer computes y = g(W u+b) with sigmoid nonlinearity, and the weights W initialized to small ran- dom Gaussian values. The last hidden layer is followed by a fully-connected layer with 10 activations (one per class) and cross-entropy loss. We trained the network for 50000 steps, with 60 examples per mini-batch. We added Batch Normalization to each hidden layer of the network, as in Sec. 3.1. We were interested in the comparison be- tween the baseline and batch-normalized networks, rather than achieving the state of the art performance on MNIST (which the described architecture does not). | 1502.03167#30 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 30,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Figure 1: (a) The test accuracy of the MNIST network the trained with and without Batch Normalization, vs. number of training steps. Batch Normalization helps the network train faster and achieve higher accuracy. (b, c) The evolution of input distributions to a typical sig- moid, over the course of training, shown as th percentiles. Batch Normalization makes the distribution more stable and reduces the internal covariate shift.\n3 fully-connected hidden layers with 100 activations each. Each hidden layer computes y = g(W u+b) with sigmoid nonlinearity, and the weights W initialized to small ran- dom Gaussian values. The last hidden layer is followed by a fully-connected layer with 10 activations (one per class) and cross-entropy loss. We trained the network for 50000 steps, with 60 examples per mini-batch. We added Batch Normalization to each hidden layer of the network, as in Sec. 3.1. We were interested in the comparison be- tween the baseline and batch-normalized networks, rather than achieving the state of the art performance on MNIST (which the described architecture does not).",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 31 | Figure 1(a) shows the fraction of correct predictions by the two networks on held-out test data, as training progresses. The batch-normalized network enjoys the higher test accuracy. To investigate why, we studied in- puts to the sigmoid, in the original network N and batch- normalized network Ntr BN (Alg. 2) over the course of train- ing. In Fig. 1(b,c) we show, for one typical activation from the last hidden layer of each network, how its distribu- tion evolves. The distributions in the original network change signiï¬cantly over time, both in their mean and the variance, which complicates the training of the sub- sequent layers. In contrast, the distributions in the batch- normalized network are much more stable as training pro- gresses, which aids the training.
# 4.2 ImageNet classiï¬cation | 1502.03167#31 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 31,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Figure 1(a) shows the fraction of correct predictions by the two networks on held-out test data, as training progresses. The batch-normalized network enjoys the higher test accuracy. To investigate why, we studied in- puts to the sigmoid, in the original network N and batch- normalized network Ntr BN (Alg. 2) over the course of train- ing. In Fig. 1(b,c) we show, for one typical activation from the last hidden layer of each network, how its distribu- tion evolves. The distributions in the original network change signiï¬cantly over time, both in their mean and the variance, which complicates the training of the sub- sequent layers. In contrast, the distributions in the batch- normalized network are much more stable as training pro- gresses, which aids the training.\n# 4.2 ImageNet classiï¬cation",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 32 | # 4.2 ImageNet classiï¬cation
We applied Batch Normalization to a new variant of the Inception network (Szegedy et al., 2014), trained on the ImageNet classiï¬cation task (Russakovsky et al., 2014). The network has a large number of convolutional and pooling layers, with a softmax layer to predict the image class, out of 1000 possibilities. Convolutional layers use ReLU as the nonlinearity. The main difference to the net- work described in (Szegedy et al., 2014) is that the 5 5 convolutional layers are replaced by two consecutive lay- 3 convolutions with up to 128 ï¬lters. The net- ers of 3 à 106 parameters, and, other than the work contains 13.6 top softmax layer, has no fully-connected layers. More
6 | 1502.03167#32 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 32,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 4.2 ImageNet classiï¬cation\nWe applied Batch Normalization to a new variant of the Inception network (Szegedy et al., 2014), trained on the ImageNet classiï¬cation task (Russakovsky et al., 2014). The network has a large number of convolutional and pooling layers, with a softmax layer to predict the image class, out of 1000 possibilities. Convolutional layers use ReLU as the nonlinearity. The main difference to the net- work described in (Szegedy et al., 2014) is that the 5 5 convolutional layers are replaced by two consecutive lay- 3 convolutions with up to 128 ï¬lters. The net- ers of 3 à 106 parameters, and, other than the work contains 13.6 top softmax layer, has no fully-connected layers. More\n6",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
1502.03167 | 33 | 6
details are given in the Appendix. We refer to this model as Inceptionin the rest of the text. The model was trained using a version of Stochastic Gradient Descent with mo- mentum (Sutskever et al., 2013), using the mini-batch size of 32. The training was performed using a large-scale, dis- tributed architecture (similar to (Dean et al., 2012)). All networks are evaluated as training progresses by comput- the probability of ing the validation accuracy @1, i.e. predicting the correct label out of 1000 possibilities, on a held-out set, using a single crop per image.
In our experiments, we evaluated several modiï¬cations of Inception with Batch Normalization. In all cases, Batch Normalization was applied to the input of each nonlinear- ity, in a convolutional way, as described in section 3.2, while keeping the rest of the architecture constant.
# 4.2.1 Accelerating BN Networks
Simply adding Batch Normalization to a network does not take full advantage of our method. To do so, we further changed the network and its training parameters, as fol- lows:
Increase learning rate. In a batch-normalized model, we have been able to achieve a training speedup from higher learning rates, with no ill side effects (Sec. 3.3). | 1502.03167#33 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 33,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "6\ndetails are given in the Appendix. We refer to this model as Inceptionin the rest of the text. The model was trained using a version of Stochastic Gradient Descent with mo- mentum (Sutskever et al., 2013), using the mini-batch size of 32. The training was performed using a large-scale, dis- tributed architecture (similar to (Dean et al., 2012)). All networks are evaluated as training progresses by comput- the probability of ing the validation accuracy @1, i.e. predicting the correct label out of 1000 possibilities, on a held-out set, using a single crop per image.\nIn our experiments, we evaluated several modiï¬cations of Inception with Batch Normalization. In all cases, Batch Normalization was applied to the input of each nonlinear- ity, in a convolutional way, as described in section 3.2, while keeping the rest of the architecture constant.\n# 4.2.1 Accelerating BN Networks\nSimply adding Batch Normalization to a network does not take full advantage of our method. To do so, we further changed the network and its training parameters, as fol- lows:\nIncrease learning rate. In a batch-normalized model, we have been able to achieve a training speedup from higher learning rates, with no ill side effects (Sec. 3.3).",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 34 | Increase learning rate. In a batch-normalized model, we have been able to achieve a training speedup from higher learning rates, with no ill side effects (Sec. 3.3).
Remove Dropout. As described in Sec. 3.4, Batch Nor- malization fulï¬lls some of the same goals as Dropout. Re- moving Dropout from Modiï¬ed BN-Inception speeds up training, without increasing overï¬tting.
Reduce the L2 weight regularization. While in Incep- tion an L2 loss on the model parameters controls overï¬t- ting, in Modiï¬ed BN-Inception the weight of this loss is reduced by a factor of 5. We ï¬nd that this improves the accuracy on the held-out validation data.
Accelerate the learning rate decay. In training Incep- tion, learning rate was decayed exponentially. Because our network trains faster than Inception, we lower the learning rate 6 times faster.
Remove Local Response Normalization While Incep- tion and other networks (Srivastava et al., 2014) beneï¬t from it, we found that with Batch Normalization it is not necessary. | 1502.03167#34 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 34,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Increase learning rate. In a batch-normalized model, we have been able to achieve a training speedup from higher learning rates, with no ill side effects (Sec. 3.3).\nRemove Dropout. As described in Sec. 3.4, Batch Nor- malization fulï¬lls some of the same goals as Dropout. Re- moving Dropout from Modiï¬ed BN-Inception speeds up training, without increasing overï¬tting.\nReduce the L2 weight regularization. While in Incep- tion an L2 loss on the model parameters controls overï¬t- ting, in Modiï¬ed BN-Inception the weight of this loss is reduced by a factor of 5. We ï¬nd that this improves the accuracy on the held-out validation data.\nAccelerate the learning rate decay. In training Incep- tion, learning rate was decayed exponentially. Because our network trains faster than Inception, we lower the learning rate 6 times faster.\nRemove Local Response Normalization While Incep- tion and other networks (Srivastava et al., 2014) beneï¬t from it, we found that with Batch Normalization it is not necessary.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 35 | Shufï¬e training examples more thoroughly. We enabled within-shard shufï¬ing of the training data, which prevents the same examples from always appearing in a mini-batch together. This led to about 1% improvements in the val- idation accuracy, which is consistent with the view of Batch Normalization as a regularizer (Sec. 3.4): the ran- domization inherent in our method should be most bene- ï¬cial when it affects an example differently each time it is seen.
Reduce the photometric distortions. Because batch- normalized networks train faster and observe each train- ing example fewer times, we let the trainer focus on more ârealâ images by distorting them less.
0.8 0.7 0.6 0.5 Inception BNâBaseline BNâx5 BNâx30 BNâx5âSigmoid Steps to match Inception 0.4 5M 10M 15M 20M 25M 30M
Figure 2: Single crop validation accuracy of Inception and its batch-normalized variants, vs. the number of training steps.
# 4.2.2 Single-Network Classiï¬cation
We evaluated the following networks, all trained on the LSVRC2012 training data, and tested on the validation data: | 1502.03167#35 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 35,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Shufï¬e training examples more thoroughly. We enabled within-shard shufï¬ing of the training data, which prevents the same examples from always appearing in a mini-batch together. This led to about 1% improvements in the val- idation accuracy, which is consistent with the view of Batch Normalization as a regularizer (Sec. 3.4): the ran- domization inherent in our method should be most bene- ï¬cial when it affects an example differently each time it is seen.\nReduce the photometric distortions. Because batch- normalized networks train faster and observe each train- ing example fewer times, we let the trainer focus on more ârealâ images by distorting them less.\n0.8 0.7 0.6 0.5 Inception BNâBaseline BNâx5 BNâx30 BNâx5âSigmoid Steps to match Inception 0.4 5M 10M 15M 20M 25M 30M \nFigure 2: Single crop validation accuracy of Inception and its batch-normalized variants, vs. the number of training steps.\n# 4.2.2 Single-Network Classiï¬cation\nWe evaluated the following networks, all trained on the LSVRC2012 training data, and tested on the validation data:",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 36 | # 4.2.2 Single-Network Classiï¬cation
We evaluated the following networks, all trained on the LSVRC2012 training data, and tested on the validation data:
Inception: the network described at the beginning of Section 4.2, trained with the initial learning rate of 0.0015. BN-Baseline: Same as Inception with Batch Normalization before each nonlinearity.
BN-x5: Inception with Batch Normalization and the modiï¬cations in Sec. 4.2.1. The initial learning rate was increased by a factor of 5, to 0.0075. The same learning rate increase with original Inception caused the model pa- rameters to reach machine inï¬nity.
BN-x30: Like BN-x5, but with the initial learning rate 0.045 (30 times that of Inception). | 1502.03167#36 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 36,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 4.2.2 Single-Network Classiï¬cation\nWe evaluated the following networks, all trained on the LSVRC2012 training data, and tested on the validation data:\nInception: the network described at the beginning of Section 4.2, trained with the initial learning rate of 0.0015. BN-Baseline: Same as Inception with Batch Normalization before each nonlinearity.\nBN-x5: Inception with Batch Normalization and the modiï¬cations in Sec. 4.2.1. The initial learning rate was increased by a factor of 5, to 0.0075. The same learning rate increase with original Inception caused the model pa- rameters to reach machine inï¬nity.\nBN-x30: Like BN-x5, but with the initial learning rate 0.045 (30 times that of Inception).",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 37 | BN-x30: Like BN-x5, but with the initial learning rate 0.045 (30 times that of Inception).
BN-x5-Sigmoid: Like BN-x5, but with sigmoid non- 1+exp(âx) instead of ReLU. We also at- linearity g(t) = tempted to train the original Inception with sigmoid, but the model remained at the accuracy equivalent to chance. In Figure 2, we show the validation accuracy of the networks, as a function of the number of training steps. 106 Inception reached the accuracy of 72.2% after 31 training steps. The Figure 3 shows, for each network, the number of training steps required to reach the same 72.2% accuracy, as well as the maximum validation accu- racy reached by the network and the number of steps to reach it. | 1502.03167#37 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 37,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "BN-x30: Like BN-x5, but with the initial learning rate 0.045 (30 times that of Inception).\nBN-x5-Sigmoid: Like BN-x5, but with sigmoid non- 1+exp(âx) instead of ReLU. We also at- linearity g(t) = tempted to train the original Inception with sigmoid, but the model remained at the accuracy equivalent to chance. In Figure 2, we show the validation accuracy of the networks, as a function of the number of training steps. 106 Inception reached the accuracy of 72.2% after 31 training steps. The Figure 3 shows, for each network, the number of training steps required to reach the same 72.2% accuracy, as well as the maximum validation accu- racy reached by the network and the number of steps to reach it.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 38 | By only using Batch Normalization (BN-Baseline), we match the accuracy of Inception in less than half the num- ber of training steps. By applying the modiï¬cations in Sec. 4.2.1, we signiï¬cantly increase the training speed of the network. BN-x5 needs 14 times fewer steps than In- Interestingly, in- ception to reach the 72.2% accuracy. creasing the learning rate further (BN-x30) causes the model to train somewhat slower initially, but allows it to 106 reach a higher ï¬nal accuracy. It reaches 74.8% after 6 steps, i.e. 5 times fewer steps than required by Inception to reach 72.2%.
We also veriï¬ed that the reduction in internal covari- ate shift allows deep networks with Batch Normalization
7
Model Inception BN-Baseline BN-x5 BN-x30 BN-x5-Sigmoid Steps to 72.2% Max accuracy 72.2% 72.7% 73.0% 74.8% 69.8% 106 106 106 106 31.0 13.3 2.1 2.7 · · · · | 1502.03167#38 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 38,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "By only using Batch Normalization (BN-Baseline), we match the accuracy of Inception in less than half the num- ber of training steps. By applying the modiï¬cations in Sec. 4.2.1, we signiï¬cantly increase the training speed of the network. BN-x5 needs 14 times fewer steps than In- Interestingly, in- ception to reach the 72.2% accuracy. creasing the learning rate further (BN-x30) causes the model to train somewhat slower initially, but allows it to 106 reach a higher ï¬nal accuracy. It reaches 74.8% after 6 steps, i.e. 5 times fewer steps than required by Inception to reach 72.2%.\nWe also veriï¬ed that the reduction in internal covari- ate shift allows deep networks with Batch Normalization\n7\nModel Inception BN-Baseline BN-x5 BN-x30 BN-x5-Sigmoid Steps to 72.2% Max accuracy 72.2% 72.7% 73.0% 74.8% 69.8% 106 106 106 106 31.0 13.3 2.1 2.7 · · · ·",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 39 | Figure 3: For Inception and the batch-normalized variants, the number of training steps required to reach the maximum accuracy of Inception (72.2%), and the maximum accuracy achieved by the net- work.
to be trained when sigmoid is used as the nonlinearity, despite the well-known difï¬culty of training such net- works. Indeed, BN-x5-Sigmoid achieves the accuracy of 69.8%. Without Batch Normalization, Inception with sig- moid never achieves better than 1/1000 accuracy.
# 4.2.3 Ensemble Classiï¬cation
The current reported best results on the ImageNet Large Scale Visual Recognition Competition are reached by the Deep Image ensemble of traditional models (Wu et al., 2015) and the ensemble model of (He et al., 2015). The latter reports the top-5 error of 4.94%, as evaluated by the ILSVRC server. Here we report a top-5 validation error of 4.9%, and test error of 4.82% (according to the ILSVRC server). This improves upon the previous best result, and exceeds the estimated accuracy of human raters according to (Russakovsky et al., 2014). | 1502.03167#39 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 39,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Figure 3: For Inception and the batch-normalized variants, the number of training steps required to reach the maximum accuracy of Inception (72.2%), and the maximum accuracy achieved by the net- work.\nto be trained when sigmoid is used as the nonlinearity, despite the well-known difï¬culty of training such net- works. Indeed, BN-x5-Sigmoid achieves the accuracy of 69.8%. Without Batch Normalization, Inception with sig- moid never achieves better than 1/1000 accuracy.\n# 4.2.3 Ensemble Classiï¬cation\nThe current reported best results on the ImageNet Large Scale Visual Recognition Competition are reached by the Deep Image ensemble of traditional models (Wu et al., 2015) and the ensemble model of (He et al., 2015). The latter reports the top-5 error of 4.94%, as evaluated by the ILSVRC server. Here we report a top-5 validation error of 4.9%, and test error of 4.82% (according to the ILSVRC server). This improves upon the previous best result, and exceeds the estimated accuracy of human raters according to (Russakovsky et al., 2014).",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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1502.03167 | 40 | For our ensemble, we used 6 networks. Each was based on BN-x30, modiï¬ed via some of the following: increased initial weights in the convolutional layers; using Dropout (with the Dropout probability of 5% or 10%, vs. 40% for the original Inception); and using non-convolutional, per-activation Batch Normalization with last hidden lay- ers of the model. Each network achieved its maximum 106 training steps. The ensemble accuracy after about 6 prediction was based on the arithmetic average of class probabilities predicted by the constituent networks. The details of ensemble and multicrop inference are similar to (Szegedy et al., 2014).
# Sep
201
We demonstrate in Fig. 4 that batch normalization al- lows us to set new state-of-the-art by a healthy margin on the ImageNet classiï¬cation challenge benchmarks.
# 5 Conclusion | 1502.03167#40 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 40,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "For our ensemble, we used 6 networks. Each was based on BN-x30, modiï¬ed via some of the following: increased initial weights in the convolutional layers; using Dropout (with the Dropout probability of 5% or 10%, vs. 40% for the original Inception); and using non-convolutional, per-activation Batch Normalization with last hidden lay- ers of the model. Each network achieved its maximum 106 training steps. The ensemble accuracy after about 6 prediction was based on the arithmetic average of class probabilities predicted by the constituent networks. The details of ensemble and multicrop inference are similar to (Szegedy et al., 2014).\n# Sep\n201\nWe demonstrate in Fig. 4 that batch normalization al- lows us to set new state-of-the-art by a healthy margin on the ImageNet classiï¬cation challenge benchmarks.\n# 5 Conclusion",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 41 | # 5 Conclusion
We have presented a novel mechanism for dramatically accelerating the training of deep networks. It is based on the premise that covariate shift, which is known to com- plicate the training of machine learning systems, also apModel GoogLeNet ensemble Deep Image low-res Deep Image high-res Deep Image ensemble BN-Inception single crop BN-Inception multicrop BN-Inception ensemble 224 256 512 variable 224 224 224 144 - - - 1 144 144 - - 24.88 - 25.2% 21.99% 20.1% | 1502.03167#41 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 41,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# 5 Conclusion\nWe have presented a novel mechanism for dramatically accelerating the training of deep networks. It is based on the premise that covariate shift, which is known to com- plicate the training of machine learning systems, also apModel GoogLeNet ensemble Deep Image low-res Deep Image high-res Deep Image ensemble BN-Inception single crop BN-Inception multicrop BN-Inception ensemble 224 256 512 variable 224 224 224 144 - - - 1 144 144 - - 24.88 - 25.2% 21.99% 20.1%",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
1502.03167 | 42 | Figure 4: Batch-Normalized Inception comparison with previous state of the art on the provided validation set com- prising 50000 images. *BN-Inception ensemble has reached 4.82% top-5 error on the 100000 images of the test set of the ImageNet as reported by the test server.
plies to sub-networks and layers, and removing it from internal activations of the network may aid in training. Our proposed method draws its power from normalizing activations, and from incorporating this normalization in the network architecture itself. This ensures that the nor- malization is appropriately handled by any optimization method that is being used to train the network. To en- able stochastic optimization methods commonly used in deep network training, we perform the normalization for each mini-batch, and backpropagate the gradients through the normalization parameters. Batch Normalization adds only two extra parameters per activation, and in doing so preserves the representation ability of the network. We presented an algorithm for constructing, training, and per- forming inference with batch-normalized networks. The resulting networks can be trained with saturating nonlin- earities, are more tolerant to increased training rates, and often do not require Dropout for regularization. | 1502.03167#42 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 42,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Figure 4: Batch-Normalized Inception comparison with previous state of the art on the provided validation set com- prising 50000 images. *BN-Inception ensemble has reached 4.82% top-5 error on the 100000 images of the test set of the ImageNet as reported by the test server.\nplies to sub-networks and layers, and removing it from internal activations of the network may aid in training. Our proposed method draws its power from normalizing activations, and from incorporating this normalization in the network architecture itself. This ensures that the nor- malization is appropriately handled by any optimization method that is being used to train the network. To en- able stochastic optimization methods commonly used in deep network training, we perform the normalization for each mini-batch, and backpropagate the gradients through the normalization parameters. Batch Normalization adds only two extra parameters per activation, and in doing so preserves the representation ability of the network. We presented an algorithm for constructing, training, and per- forming inference with batch-normalized networks. The resulting networks can be trained with saturating nonlin- earities, are more tolerant to increased training rates, and often do not require Dropout for regularization.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 43 | Merely adding Batch Normalization to a state-of-the- art image classiï¬cation model yields a substantial speedup in training. By further increasing the learning rates, re- moving Dropout, and applying other modiï¬cations af- forded by Batch Normalization, we reach the previous state of the art with only a small fraction of training steps â and then beat the state of the art in single-network image classiï¬cation. Furthermore, by combining multiple mod- els trained with Batch Normalization, we perform better than the best known system on ImageNet, by a signiï¬cant margin. | 1502.03167#43 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 43,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Merely adding Batch Normalization to a state-of-the- art image classiï¬cation model yields a substantial speedup in training. By further increasing the learning rates, re- moving Dropout, and applying other modiï¬cations af- forded by Batch Normalization, we reach the previous state of the art with only a small fraction of training steps â and then beat the state of the art in single-network image classiï¬cation. Furthermore, by combining multiple mod- els trained with Batch Normalization, we perform better than the best known system on ImageNet, by a signiï¬cant margin.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 44 | Interestingly, our method bears similarity to the stan- dardization layer of (G¨ulc¸ehre & Bengio, 2013), though the two methods stem from very different goals, and per- form different tasks. The goal of Batch Normalization is to achieve a stable distribution of activation values throughout training, and in our experiments we apply it before the nonlinearity since that is where matching the ï¬rst and second moments is more likely to result in a stable distribution. On the contrary, (G¨ulc¸ehre & Bengio, 2013) apply the standardization layer to the output of the nonlinearity, which results in sparser activations. In our large-scale image classiï¬cation experiments, we have not observed the nonlinearity inputs to be sparse, neither with nor without Batch Normalization. Other notable differentiating characteristics of Batch Normalization include the learned scale and shift that allow the BN transform to represent identity (the standardization layer did not re- quire this since it was followed by the learned linear trans- form that, conceptually, absorbs the necessary scale and shift), handling of convolutional layers, deterministic in- ference that does not depend on the mini-batch, and batch- normalizing each convolutional layer in the network. | 1502.03167#44 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 44,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Interestingly, our method bears similarity to the stan- dardization layer of (G¨ulc¸ehre & Bengio, 2013), though the two methods stem from very different goals, and per- form different tasks. The goal of Batch Normalization is to achieve a stable distribution of activation values throughout training, and in our experiments we apply it before the nonlinearity since that is where matching the ï¬rst and second moments is more likely to result in a stable distribution. On the contrary, (G¨ulc¸ehre & Bengio, 2013) apply the standardization layer to the output of the nonlinearity, which results in sparser activations. In our large-scale image classiï¬cation experiments, we have not observed the nonlinearity inputs to be sparse, neither with nor without Batch Normalization. Other notable differentiating characteristics of Batch Normalization include the learned scale and shift that allow the BN transform to represent identity (the standardization layer did not re- quire this since it was followed by the learned linear trans- form that, conceptually, absorbs the necessary scale and shift), handling of convolutional layers, deterministic in- ference that does not depend on the mini-batch, and batch- normalizing each convolutional layer in the network.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 45 | In this work, we have not explored the full range of possibilities that Batch Normalization potentially enables. Our future work includes applications of our method to Recurrent Neural Networks (Pascanu et al., 2013), where the internal covariate shift and the vanishing or exploding gradients may be especially severe, and which would al- low us to more thoroughly test the hypothesis that normal- ization improves gradient propagation (Sec. 3.3). We plan to investigate whether Batch Normalization can help with domain adaptation, in its traditional sense â i.e. whether the normalization performed by the network would al- low it to more easily generalize to new data distribu- tions, perhaps with just a recomputation of the population means and variances (Alg. 2). Finally, we believe that fur- ther theoretical analysis of the algorithm would allow still more improvements and applications.
# References
Bengio, Yoshua and Glorot, Xavier. Understanding the difï¬culty of training deep feedforward neural networks. In Proceedings of AISTATS 2010, volume 9, pp. 249â 256, May 2010. | 1502.03167#45 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 45,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "In this work, we have not explored the full range of possibilities that Batch Normalization potentially enables. Our future work includes applications of our method to Recurrent Neural Networks (Pascanu et al., 2013), where the internal covariate shift and the vanishing or exploding gradients may be especially severe, and which would al- low us to more thoroughly test the hypothesis that normal- ization improves gradient propagation (Sec. 3.3). We plan to investigate whether Batch Normalization can help with domain adaptation, in its traditional sense â i.e. whether the normalization performed by the network would al- low it to more easily generalize to new data distribu- tions, perhaps with just a recomputation of the population means and variances (Alg. 2). Finally, we believe that fur- ther theoretical analysis of the algorithm would allow still more improvements and applications.\n# References\nBengio, Yoshua and Glorot, Xavier. Understanding the difï¬culty of training deep feedforward neural networks. In Proceedings of AISTATS 2010, volume 9, pp. 249â 256, May 2010.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 46 | Dean, Jeffrey, Corrado, Greg S., Monga, Rajat, Chen, Kai, Devin, Matthieu, Le, Quoc V., Mao, Mark Z., Ranzato, MarcâAurelio, Senior, Andrew, Tucker, Paul, Yang, Ke, and Ng, Andrew Y. Large scale distributed deep net- works. In NIPS, 2012.
Desjardins, Guillaume and Kavukcuoglu, Koray. Natural neural networks. (unpublished).
Duchi, John, Hazan, Elad, and Singer, Yoram. Adaptive subgradient methods for online learning and stochastic
8
optimization. J. Mach. Learn. Res., 12:2121â2159, July 2011. ISSN 1532-4435.
G¨ulc¸ehre, C¸ aglar and Bengio, Yoshua. Knowledge mat- ters: Importance of prior information for optimization. CoRR, abs/1301.4083, 2013.
He, K., Zhang, X., Ren, S., and Sun, J. Delving Deep into Rectiï¬ers: Surpassing Human-Level Performance on ImageNet Classiï¬cation. ArXiv e-prints, February 2015. | 1502.03167#46 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 46,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Dean, Jeffrey, Corrado, Greg S., Monga, Rajat, Chen, Kai, Devin, Matthieu, Le, Quoc V., Mao, Mark Z., Ranzato, MarcâAurelio, Senior, Andrew, Tucker, Paul, Yang, Ke, and Ng, Andrew Y. Large scale distributed deep net- works. In NIPS, 2012.\nDesjardins, Guillaume and Kavukcuoglu, Koray. Natural neural networks. (unpublished).\nDuchi, John, Hazan, Elad, and Singer, Yoram. Adaptive subgradient methods for online learning and stochastic\n8\noptimization. J. Mach. Learn. Res., 12:2121â2159, July 2011. ISSN 1532-4435.\nG¨ulc¸ehre, C¸ aglar and Bengio, Yoshua. Knowledge mat- ters: Importance of prior information for optimization. CoRR, abs/1301.4083, 2013.\nHe, K., Zhang, X., Ren, S., and Sun, J. Delving Deep into Rectiï¬ers: Surpassing Human-Level Performance on ImageNet Classiï¬cation. ArXiv e-prints, February 2015.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 47 | Hyv¨arinen, A. and Oja, E. Independent component anal- ysis: Algorithms and applications. Neural Netw., 13 (4-5):411â430, May 2000.
Jiang, Jing. A literature survey on domain adaptation of statistical classiï¬ers, 2008.
LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recog- nition. Proceedings of the IEEE, 86(11):2278â2324, November 1998a.
LeCun, Y., Bottou, L., Orr, G., and Muller, K. Efï¬cient backprop. In Orr, G. and K., Muller (eds.), Neural Net- works: Tricks of the trade. Springer, 1998b.
Lyu, S and Simoncelli, E P. Nonlinear image representa- tion using divisive normalization. In Proc. Computer Vision and Pattern Recognition, pp. 1â8. IEEE Com- puter Society, Jun 23-28 2008. doi: 10.1109/CVPR. 2008.4587821.
Nair, Vinod and Hinton, Geoffrey E. Rectiï¬ed linear units improve restricted boltzmann machines. In ICML, pp. 807â814. Omnipress, 2010. | 1502.03167#47 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 47,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Hyv¨arinen, A. and Oja, E. Independent component anal- ysis: Algorithms and applications. Neural Netw., 13 (4-5):411â430, May 2000.\nJiang, Jing. A literature survey on domain adaptation of statistical classiï¬ers, 2008.\nLeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recog- nition. Proceedings of the IEEE, 86(11):2278â2324, November 1998a.\nLeCun, Y., Bottou, L., Orr, G., and Muller, K. Efï¬cient backprop. In Orr, G. and K., Muller (eds.), Neural Net- works: Tricks of the trade. Springer, 1998b.\nLyu, S and Simoncelli, E P. Nonlinear image representa- tion using divisive normalization. In Proc. Computer Vision and Pattern Recognition, pp. 1â8. IEEE Com- puter Society, Jun 23-28 2008. doi: 10.1109/CVPR. 2008.4587821.\nNair, Vinod and Hinton, Geoffrey E. Rectiï¬ed linear units improve restricted boltzmann machines. In ICML, pp. 807â814. Omnipress, 2010.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 48 | Pascanu, Razvan, Mikolov, Tomas, and Bengio, Yoshua. On the difï¬culty of training recurrent neural networks. In Proceedings of the 30th International Conference on Machine Learning, ICML 2013, Atlanta, GA, USA, 16- 21 June 2013, pp. 1310â1318, 2013.
Povey, Daniel, Zhang, Xiaohui, and Khudanpur, San- jeev. Parallel training of deep neural networks with CoRR, natural gradient and parameter averaging. abs/1410.7455, 2014.
Raiko, Tapani, Valpola, Harri, and LeCun, Yann. Deep learning made easier by linear transformations in per- ceptrons. In International Conference on Artiï¬cial In- telligence and Statistics (AISTATS), pp. 924â932, 2012.
Russakovsky, Olga, Deng, Jia, Su, Hao, Krause, Jonathan, Satheesh, Sanjeev, Ma, Sean, Huang, Zhiheng, Karpa- thy, Andrej, Khosla, Aditya, Bernstein, Michael, Berg, Alexander C., and Fei-Fei, Li. ImageNet Large Scale Visual Recognition Challenge, 2014.
9 | 1502.03167#48 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 48,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Pascanu, Razvan, Mikolov, Tomas, and Bengio, Yoshua. On the difï¬culty of training recurrent neural networks. In Proceedings of the 30th International Conference on Machine Learning, ICML 2013, Atlanta, GA, USA, 16- 21 June 2013, pp. 1310â1318, 2013.\nPovey, Daniel, Zhang, Xiaohui, and Khudanpur, San- jeev. Parallel training of deep neural networks with CoRR, natural gradient and parameter averaging. abs/1410.7455, 2014.\nRaiko, Tapani, Valpola, Harri, and LeCun, Yann. Deep learning made easier by linear transformations in per- ceptrons. In International Conference on Artiï¬cial In- telligence and Statistics (AISTATS), pp. 924â932, 2012.\nRussakovsky, Olga, Deng, Jia, Su, Hao, Krause, Jonathan, Satheesh, Sanjeev, Ma, Sean, Huang, Zhiheng, Karpa- thy, Andrej, Khosla, Aditya, Bernstein, Michael, Berg, Alexander C., and Fei-Fei, Li. ImageNet Large Scale Visual Recognition Challenge, 2014.\n9",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 49 | 9
Saxe, Andrew M., McClelland, James L., and Ganguli, Surya. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. CoRR, abs/1312.6120, 2013.
Improving predictive inference under covariate shift by weighting the log-likelihood function. Journal of Statistical Planning and Inference, 90(2):227â244, October 2000.
Srivastava, Nitish, Hinton, Geoffrey, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: A simple way to prevent neural networks from overï¬t- ting. J. Mach. Learn. Res., 15(1):1929â1958, January 2014.
Sutskever, Ilya, Martens, James, Dahl, George E., and Hinton, Geoffrey E. On the importance of initial- ization and momentum in deep learning. In ICML (3), volume 28 of JMLR Proceedings, pp. 1139â1147. JMLR.org, 2013. | 1502.03167#49 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 49,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "9\nSaxe, Andrew M., McClelland, James L., and Ganguli, Surya. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. CoRR, abs/1312.6120, 2013.\nImproving predictive inference under covariate shift by weighting the log-likelihood function. Journal of Statistical Planning and Inference, 90(2):227â244, October 2000.\nSrivastava, Nitish, Hinton, Geoffrey, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: A simple way to prevent neural networks from overï¬t- ting. J. Mach. Learn. Res., 15(1):1929â1958, January 2014.\nSutskever, Ilya, Martens, James, Dahl, George E., and Hinton, Geoffrey E. On the importance of initial- ization and momentum in deep learning. In ICML (3), volume 28 of JMLR Proceedings, pp. 1139â1147. JMLR.org, 2013.",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
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# Appendix
# Variant of the Inception Model Used | 1502.03167#50 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 50,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "Szegedy, Christian, Liu, Wei, Jia, Yangqing, Sermanet, Pierre, Reed, Scott, Anguelov, Dragomir, Erhan, Du- mitru, Vanhoucke, Vincent, and Rabinovich, An- CoRR, Going deeper with convolutions. drew. abs/1409.4842, 2014.\nWiesler, Simon and Ney, Hermann. A convergence anal- ysis of log-linear training. In Shawe-Taylor, J., Zemel, R.S., Bartlett, P., Pereira, F.C.N., and Weinberger, K.Q. (eds.), Advances in Neural Information Processing Sys- tems 24, pp. 657â665, Granada, Spain, December 2011.\nWiesler, Simon, Richard, Alexander, Schl¨uter, Ralf, and Ney, Hermann. Mean-normalized stochastic gradient for large-scale deep learning. In IEEE International Conference on Acoustics, Speech, and Signal Process- ing, pp. 180â184, Florence, Italy, May 2014.\nWu, Ren, Yan, Shengen, Shan, Yi, Dang, Qingqing, and Sun, Gang. Deep image: Scaling up image recognition, 2015.\n# Appendix\n# Variant of the Inception Model Used",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
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] |
1502.03167 | 51 | # Appendix
# Variant of the Inception Model Used
Figure 5 documents the changes that were performed compared to the architecture with respect to the GoogleNet archictecture. For the interpretation of this table, please consult (Szegedy et al., 2014). The notable architecture changes compared to the GoogLeNet model include:
5 convolutional layers are replaced by two The 5 Ã consecutive 3 This in- creases the maximum depth of the network by 9
weight layers. Also it increases the number of pa- rameters by 25% and the computational cost is in- creased by about 30%.
⢠The number 28 from 2 to 3. à 28 inception modules is increased
Inside the modules, sometimes average, sometimes maximum-pooling is employed. This is indicated in the entries corresponding to the pooling layers of the table.
There are no across the board pooling layers be- tween any two Inception modules, but stride-2 con- volution/pooling layers are employed before the ï¬l- ter concatenation in the modules 3c, 4e.
Our model employed separable convolution with depth multiplier 8 on the ï¬rst convolutional layer. This reduces the computational cost while increasing the memory con- sumption at training time.
10 | 1502.03167#51 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 51,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "# Appendix\n# Variant of the Inception Model Used\nFigure 5 documents the changes that were performed compared to the architecture with respect to the GoogleNet archictecture. For the interpretation of this table, please consult (Szegedy et al., 2014). The notable architecture changes compared to the GoogLeNet model include:\n5 convolutional layers are replaced by two The 5 à consecutive 3 This in- creases the maximum depth of the network by 9\nweight layers. Also it increases the number of pa- rameters by 25% and the computational cost is in- creased by about 30%.\n⢠The number 28 from 2 to 3. à 28 inception modules is increased\nInside the modules, sometimes average, sometimes maximum-pooling is employed. This is indicated in the entries corresponding to the pooling layers of the table.\nThere are no across the board pooling layers be- tween any two Inception modules, but stride-2 con- volution/pooling layers are employed before the ï¬l- ter concatenation in the modules 3c, 4e.\nOur model employed separable convolution with depth multiplier 8 on the ï¬rst convolutional layer. This reduces the computational cost while increasing the memory con- sumption at training time.\n10",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.03167 | 52 | #3Ã3 reduce double #3Ã3 reduce patch size/ stride 7Ã7/2 3Ã3/2 3Ã3/1 3Ã3/2 output size 112Ã112Ã64 56Ã56Ã64 56Ã56Ã192 28Ã28Ã192 28Ã28Ã256 28Ã28Ã320 28Ã28Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã1024 7Ã7Ã1024 7Ã7Ã1024 1Ã1Ã1024 double #3Ã3 #3Ã3 depth #1Ã1 Pool +proj type convolution* max pool convolution max pool inception (3a) inception (3b) inception (3c) inception (4a) inception (4b) inception (4c) inception (4d) inception (4e) inception (5a) inception (5b) avg pool 1 0 1 0 3 3 3 3 3 3 3 3 3 3 0 192 64 64 96 160 96 128 160 192 192 320 320 64 64 0 224 192 160 96 0 352 352 64 64 64 96 96 128 160 192 160 192 64 64 128 64 96 128 128 128 192 192 96 96 96 128 128 160 192 256 224 224 avg + 32 avg + 64 max + pass through avg + 128 avg + 128 avg + 128 avg | 1502.03167#52 | Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift | Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. Batch Normalization allows us to use much higher learning rates and
be less careful about initialization. It also acts as a regularizer, in some
cases eliminating the need for Dropout. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters. | http://arxiv.org/pdf/1502.03167 | Sergey Ioffe, Christian Szegedy | cs.LG | null | null | cs.LG | 20150211 | 20150302 | [
{
"id": "1502.03167"
}
] | {
"authors": "Sergey Ioffe, Christian Szegedy",
"chunk_id": 52,
"doc_id": "1502.03167",
"primary_category": "cs.LG",
"published": 20150211,
"source": "http://arxiv.org/pdf/1502.03167",
"summary": "Training Deep Neural Networks is complicated by the fact that the\ndistribution of each layer's inputs changes during training, as the parameters\nof the previous layers change. This slows down the training by requiring lower\nlearning rates and careful parameter initialization, and makes it notoriously\nhard to train models with saturating nonlinearities. We refer to this\nphenomenon as internal covariate shift, and address the problem by normalizing\nlayer inputs. Our method draws its strength from making normalization a part of\nthe model architecture and performing the normalization for each training\nmini-batch. Batch Normalization allows us to use much higher learning rates and\nbe less careful about initialization. It also acts as a regularizer, in some\ncases eliminating the need for Dropout. Applied to a state-of-the-art image\nclassification model, Batch Normalization achieves the same accuracy with 14\ntimes fewer training steps, and beats the original model by a significant\nmargin. Using an ensemble of batch-normalized networks, we improve upon the\nbest published result on ImageNet classification: reaching 4.9% top-5\nvalidation error (and 4.8% test error), exceeding the accuracy of human raters.",
"text": "#3Ã3 reduce double #3Ã3 reduce patch size/ stride 7Ã7/2 3Ã3/2 3Ã3/1 3Ã3/2 output size 112Ã112Ã64 56Ã56Ã64 56Ã56Ã192 28Ã28Ã192 28Ã28Ã256 28Ã28Ã320 28Ã28Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã576 14Ã14Ã1024 7Ã7Ã1024 7Ã7Ã1024 1Ã1Ã1024 double #3Ã3 #3Ã3 depth #1Ã1 Pool +proj type convolution* max pool convolution max pool inception (3a) inception (3b) inception (3c) inception (4a) inception (4b) inception (4c) inception (4d) inception (4e) inception (5a) inception (5b) avg pool 1 0 1 0 3 3 3 3 3 3 3 3 3 3 0 192 64 64 96 160 96 128 160 192 192 320 320 64 64 0 224 192 160 96 0 352 352 64 64 64 96 96 128 160 192 160 192 64 64 128 64 96 128 128 128 192 192 96 96 96 128 128 160 192 256 224 224 avg + 32 avg + 64 max + pass through avg + 128 avg + 128 avg + 128 avg",
"title": "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift",
"year": 2015
} | [
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] |
1502.02251 | 0 | 5 1 0 2
n u J 8 1 ] L M . t a t s [
3 v 1 5 2 2 0 . 2 0 5 1 : v i X r a
# From Pixels to Torques: Policy Learning with Deep Dynamical Models
# Niklas Wahlstr¨om Division of Automatic Control, Link¨oping University, Link¨oping, Sweden
[email protected]
# Thomas B. Sch¨on Department of Information Technology, Uppsala University, Sweden
# [email protected]
# Marc Peter Deisenroth Department of Computing, Imperial College London, United Kingdom
[email protected]
# Abstract | 1502.02251#0 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 0,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "5 1 0 2\nn u J 8 1 ] L M . t a t s [\n3 v 1 5 2 2 0 . 2 0 5 1 : v i X r a\n# From Pixels to Torques: Policy Learning with Deep Dynamical Models\n# Niklas Wahlstr¨om Division of Automatic Control, Link¨oping University, Link¨oping, Sweden\[email protected]\n# Thomas B. Sch¨on Department of Information Technology, Uppsala University, Sweden\n# [email protected]\n# Marc Peter Deisenroth Department of Computing, Imperial College London, United Kingdom\[email protected]\n# Abstract",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 1 | # Marc Peter Deisenroth Department of Computing, Imperial College London, United Kingdom
[email protected]
# Abstract
Data-efï¬cient learning in continuous state-action spaces using very high-dimensional observations remains a key challenge in developing fully autonomous systems. In this paper, we con- the pix- sider one instance of this challenge, els to torques problem, where an agent must learn a closed-loop control policy from pixel in- formation only. We introduce a data-efï¬cient, model-based reinforcement learning algorithm that learns such a closed-loop policy directly from pixel information. The key ingredient is a deep dynamical model that uses deep auto- encoders to learn a low-dimensional embedding of images jointly with a predictive model in this low-dimensional feature space. Joint learning ensures that not only static but also dynamic properties of the data are accounted for. This is crucial for long-term predictions, which lie at the core of the adaptive model predictive con- trol strategy that we use for closed-loop con- trol. Compared to state-of-the-art reinforcement learning methods for continuous states and ac- tions, our approach learns quickly, scales to high- dimensional state spaces and is an important step toward fully autonomous learning from pixels to torques. | 1502.02251#1 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 1,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# Marc Peter Deisenroth Department of Computing, Imperial College London, United Kingdom\[email protected]\n# Abstract\nData-efï¬cient learning in continuous state-action spaces using very high-dimensional observations remains a key challenge in developing fully autonomous systems. In this paper, we con- the pix- sider one instance of this challenge, els to torques problem, where an agent must learn a closed-loop control policy from pixel in- formation only. We introduce a data-efï¬cient, model-based reinforcement learning algorithm that learns such a closed-loop policy directly from pixel information. The key ingredient is a deep dynamical model that uses deep auto- encoders to learn a low-dimensional embedding of images jointly with a predictive model in this low-dimensional feature space. Joint learning ensures that not only static but also dynamic properties of the data are accounted for. This is crucial for long-term predictions, which lie at the core of the adaptive model predictive con- trol strategy that we use for closed-loop con- trol. Compared to state-of-the-art reinforcement learning methods for continuous states and ac- tions, our approach learns quickly, scales to high- dimensional state spaces and is an important step toward fully autonomous learning from pixels to torques.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 2 | mation, (3) take new information into account for learning and adaptation. Effectively, any fully autonomous system has to close this perception-action-learning loop without relying on speciï¬c human expert knowledge. The pixels to torques problem (Brock, 2011) identiï¬es key aspects of an autonomous system: autonomous thinking and decision making using sensor measurements only, intelligent explo- ration and learning from mistakes.
We consider the problem of learning closed-loop policies (âtorquesâ) from pixel information end-to-end. A possible scenario is a scene in which a robot is moving about. The only available sensor information is provided by a camera, i.e., no direct information of the robotâs joint conï¬gura- tion is available. The objective is to learn a continuous- valued policy that allows the robotic agent to solve a task in this continuous environment in a data-efï¬cient way, i.e., we want to keep the number of trials small. To date, there is no fully autonomous system that convincingly closes the perception-action-learning loop and solves the pixels to torques problem in continuous state-action spaces, the natural domains in robotics. | 1502.02251#2 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 2,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "mation, (3) take new information into account for learning and adaptation. Effectively, any fully autonomous system has to close this perception-action-learning loop without relying on speciï¬c human expert knowledge. The pixels to torques problem (Brock, 2011) identiï¬es key aspects of an autonomous system: autonomous thinking and decision making using sensor measurements only, intelligent explo- ration and learning from mistakes.\nWe consider the problem of learning closed-loop policies (âtorquesâ) from pixel information end-to-end. A possible scenario is a scene in which a robot is moving about. The only available sensor information is provided by a camera, i.e., no direct information of the robotâs joint conï¬gura- tion is available. The objective is to learn a continuous- valued policy that allows the robotic agent to solve a task in this continuous environment in a data-efï¬cient way, i.e., we want to keep the number of trials small. To date, there is no fully autonomous system that convincingly closes the perception-action-learning loop and solves the pixels to torques problem in continuous state-action spaces, the natural domains in robotics.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 3 | A promising approach toward solving the pixels to torques problem is Reinforcement Learning (RL) (Sutton & Barto, 1998), a principled mathematical framework that deals with fully autonomous learning from trial and error. How- ever, one practical shortcoming of many existing RL algo- rithms is that they require many trials to learn good poli- cies, which is prohibitive when working with real-world mechanical plants or robots.
# 1. Introduction
The vision of fully autonomous and intelligent systems that learn by themselves has inï¬uenced AI and robotics re- search for many decades. To devise fully autonomous sys- tems, it is necessary to (1) process perceptual data (e.g., im- ages) to summarize knowledge about the surrounding envi- ronment and the systemâs behavior in this environment, (2) make decisions based on uncertain and incomplete inforOne way of using data efï¬ciently (and therefore keep the number of experiments small) is to learn forward models of the underlying dynamical system, which are then used for internal simulations and policy learning. These ideas have been successfully applied to RL, control and robotics in (Schmidhuber, 1990; Atkeson & Schaal, 1997; Bagnell & Schneider, 2001; Contardo et al., 2013;
From Pixels to Torques: Policy Learning with Deep Dynamical Models | 1502.02251#3 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 3,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "A promising approach toward solving the pixels to torques problem is Reinforcement Learning (RL) (Sutton & Barto, 1998), a principled mathematical framework that deals with fully autonomous learning from trial and error. How- ever, one practical shortcoming of many existing RL algo- rithms is that they require many trials to learn good poli- cies, which is prohibitive when working with real-world mechanical plants or robots.\n# 1. Introduction\nThe vision of fully autonomous and intelligent systems that learn by themselves has inï¬uenced AI and robotics re- search for many decades. To devise fully autonomous sys- tems, it is necessary to (1) process perceptual data (e.g., im- ages) to summarize knowledge about the surrounding envi- ronment and the systemâs behavior in this environment, (2) make decisions based on uncertain and incomplete inforOne way of using data efï¬ciently (and therefore keep the number of experiments small) is to learn forward models of the underlying dynamical system, which are then used for internal simulations and policy learning. These ideas have been successfully applied to RL, control and robotics in (Schmidhuber, 1990; Atkeson & Schaal, 1997; Bagnell & Schneider, 2001; Contardo et al., 2013;\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 4 | From Pixels to Torques: Policy Learning with Deep Dynamical Models
Image at time t-1 Vr-1 Zr] Feature at time t-1 Prediction model Encoder ââ__> âââ> g! Decoder eS g Feature at time t Image at time t Zt vr
Figure 1. Illustration of our idea of combining deep learning architectures for feature learning and prediction models in feature space. A camera observes a robot approaching an object. A good low-dimensional feature representation of an image is important for learning a predictive model if the camera is the only sensor available.
Pan & Theodorou, 2014; Deisenroth et al., 2015; Pan & Theodorou, 2014; van Hoof et al., 2015; Levine et al., 2015), for instance. However, these methods use heuris- tic or engineered low-dimensional features, and they do not easily scale to data-efï¬cient RL using pixel informa- tion only because even âsmallâ images possess thousands of dimensions. | 1502.02251#4 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 4,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "From Pixels to Torques: Policy Learning with Deep Dynamical Models\nImage at time t-1 Vr-1 Zr] Feature at time t-1 Prediction model Encoder ââ__> âââ> g! Decoder eS g Feature at time t Image at time t Zt vr\nFigure 1. Illustration of our idea of combining deep learning architectures for feature learning and prediction models in feature space. A camera observes a robot approaching an object. A good low-dimensional feature representation of an image is important for learning a predictive model if the camera is the only sensor available.\nPan & Theodorou, 2014; Deisenroth et al., 2015; Pan & Theodorou, 2014; van Hoof et al., 2015; Levine et al., 2015), for instance. However, these methods use heuris- tic or engineered low-dimensional features, and they do not easily scale to data-efï¬cient RL using pixel informa- tion only because even âsmallâ images possess thousands of dimensions.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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1502.02251 | 5 | we can use for internal simulation of the dynamical sys- tem. For this purpose, we employ deep auto-encoders for the lower-dimensional embedding and a multi-layer feed- forward neural network for the transition function. We use this deep dynamical model to predict trajectories and apply an adaptive model-predictive-control (MPC) algo- rithm (Mayne, 2014) for online closed-loop control, which is practically based on pixel information only.
A common way of dealing with high-dimensional data is to learn low-dimensional feature representations. Deep learn- ing architectures, such as deep neural networks (Hinton & Salakhutdinov, 2006), stacked auto-encoders (Bengio et al., 2007; Vincent et al., 2008), or convolutional neu- ral networks (LeCun et al., 1998), are the current state of the art in learning parsimonious representations of high- dimensional data. Deep learning has been successfully ap- plied to image, text and speech data in commercial prod- ucts, e.g., by Google, Amazon and Facebook. | 1502.02251#5 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 5,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "we can use for internal simulation of the dynamical sys- tem. For this purpose, we employ deep auto-encoders for the lower-dimensional embedding and a multi-layer feed- forward neural network for the transition function. We use this deep dynamical model to predict trajectories and apply an adaptive model-predictive-control (MPC) algo- rithm (Mayne, 2014) for online closed-loop control, which is practically based on pixel information only.\nA common way of dealing with high-dimensional data is to learn low-dimensional feature representations. Deep learn- ing architectures, such as deep neural networks (Hinton & Salakhutdinov, 2006), stacked auto-encoders (Bengio et al., 2007; Vincent et al., 2008), or convolutional neu- ral networks (LeCun et al., 1998), are the current state of the art in learning parsimonious representations of high- dimensional data. Deep learning has been successfully ap- plied to image, text and speech data in commercial prod- ucts, e.g., by Google, Amazon and Facebook.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 6 | Deep learning has been used to produce ï¬rst promising results in the context of model-free RL on images: For instance, (Mnih et al., 2015) present an approach based on Deep-Q-learning, in which human-level game strategies are learned autonomously, purely based on pixel informa- tion. Moreover, (Lange et al., 2012) presented an approach that learns good discrete actions to control a slot car based on raw images, employing deep architectures for ï¬nding compact low-dimensional representations. Other examples of deep learning in the context of RL on image data in- clude (Cuccu et al., 2011; Koutnik et al., 2013). These ap- proaches have in common that they try to estimate the value function from which the policy is derived. However, nei- ther of these algorithms learns a predictive model and are, therefore, prone to data inefï¬ciency, either requiring data collection from millions of experiments or relying on dis- cretization and very low-dimensional feature spaces, limit- ing their applicability to mechanical systems. | 1502.02251#6 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 6,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Deep learning has been used to produce ï¬rst promising results in the context of model-free RL on images: For instance, (Mnih et al., 2015) present an approach based on Deep-Q-learning, in which human-level game strategies are learned autonomously, purely based on pixel informa- tion. Moreover, (Lange et al., 2012) presented an approach that learns good discrete actions to control a slot car based on raw images, employing deep architectures for ï¬nding compact low-dimensional representations. Other examples of deep learning in the context of RL on image data in- clude (Cuccu et al., 2011; Koutnik et al., 2013). These ap- proaches have in common that they try to estimate the value function from which the policy is derived. However, nei- ther of these algorithms learns a predictive model and are, therefore, prone to data inefï¬ciency, either requiring data collection from millions of experiments or relying on dis- cretization and very low-dimensional feature spaces, limit- ing their applicability to mechanical systems.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 7 | To increase data efï¬ciency, we therefore introduce a model- based approach to learning from pixels to torques. In par- ticular, exploit results from (Wahlstr¨om et al., 2015) and jointly learn a lower-dimensional embedding of images and a transition function in this lower-dimensional space that
MPC has been well explored in the control community, However, adaptive MPC has so far not received much atten- tion in the literature (Mayne, 2014). An exception is (Sha, 2008), where the authors advocate a neural network ap- proach similar to ours. However, they do not consider high- dimensional data but assume that they have direct access to low-dimensional measurements.
Our approach beneï¬ts from the application of model- based optimal control principles within a machine learn- ing framework. Along these lines, (Deisenroth et al., 2009; Abramova et al., 2012; Boedecker et al., 2014; Pan & Theodorou, 2014; Levine et al., 2015) suggested to ï¬rst learn a transition model and then use optimal control meth- ods to solve RL problems. Unlike these methods, our ap- proach does not need to estimate value functions and scales to high-dimensional problems. | 1502.02251#7 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 7,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "To increase data efï¬ciency, we therefore introduce a model- based approach to learning from pixels to torques. In par- ticular, exploit results from (Wahlstr¨om et al., 2015) and jointly learn a lower-dimensional embedding of images and a transition function in this lower-dimensional space that\nMPC has been well explored in the control community, However, adaptive MPC has so far not received much atten- tion in the literature (Mayne, 2014). An exception is (Sha, 2008), where the authors advocate a neural network ap- proach similar to ours. However, they do not consider high- dimensional data but assume that they have direct access to low-dimensional measurements.\nOur approach beneï¬ts from the application of model- based optimal control principles within a machine learn- ing framework. Along these lines, (Deisenroth et al., 2009; Abramova et al., 2012; Boedecker et al., 2014; Pan & Theodorou, 2014; Levine et al., 2015) suggested to ï¬rst learn a transition model and then use optimal control meth- ods to solve RL problems. Unlike these methods, our ap- proach does not need to estimate value functions and scales to high-dimensional problems.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 8 | Similar to our approach, (Boots et al., 2014; Levine et al., 2015; van Hoof et al., 2015) recently proposed model- based RL methods that learn policies directly from vi- sual information. Unlike these methods, we exploit a low- dimensional feature representation that allows for fast pre- dictions and online control learning via MPC.
# Problem Set-up and Objective
We consider a classical N-step finite-horizon RL setting in which an agent attempts to solve a particular task by trial and error. In particular, our objective is to find a closed-loop policy 7* that minimizes the long-term cost v= yea fo(xt, uz), where fo denotes an immediate cost, 7, ⬠R? is the continuous-valued system state and uz ⬠RF are continuous control inputs.
â
From Pixels to Torques: Policy Learning with Deep Dynamical Models
Input layer (high-dim. data) Y1yt Hidden layer (feature) Output layer (reconstructed) YL Encoder g~1 Decoder g | 1502.02251#8 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 8,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Similar to our approach, (Boots et al., 2014; Levine et al., 2015; van Hoof et al., 2015) recently proposed model- based RL methods that learn policies directly from vi- sual information. Unlike these methods, we exploit a low- dimensional feature representation that allows for fast pre- dictions and online control learning via MPC.\n# Problem Set-up and Objective\nWe consider a classical N-step finite-horizon RL setting in which an agent attempts to solve a particular task by trial and error. In particular, our objective is to find a closed-loop policy 7* that minimizes the long-term cost v= yea fo(xt, uz), where fo denotes an immediate cost, 7, ⬠R? is the continuous-valued system state and uz ⬠RF are continuous control inputs.\nâ\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\nInput layer (high-dim. data) Y1yt Hidden layer (feature) Output layer (reconstructed) YL Encoder g~1 Decoder g",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 9 | Input layer (high-dim. data) Y1yt Hidden layer (feature) Output layer (reconstructed) YL Encoder g~1 Decoder g
Figure 2. Auto-encoder that consists of an encoder g~! and a decoder g. The encoder maps the original image yw ⬠R⢠onto its low-dimensional representation z; = goâ (ys) eRâ, where m < M; the decoder maps this feature back to a high- dimensional representation 7 = g(Z). The gray color represents high-dimensional observations.
High-dim. observations Features Control inputs
Figure 3. Prediction model: Each feature z; is computed from high-dimensional data y; via the encoder g~'. The transition model predicts the feature 2,41)/,,, at the next time step based on the n-step history of n past features z;-n41,..., Z¢ and con- trol inputs weân+1,-.., ut. The predicted feature 241), can be mapped to a high-dimensional prediction #41 via the decoder g. The gray color represents high-dimensional observations.
# 2.1. Deep Auto-Encoder | 1502.02251#9 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 9,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Input layer (high-dim. data) Y1yt Hidden layer (feature) Output layer (reconstructed) YL Encoder g~1 Decoder g\nFigure 2. Auto-encoder that consists of an encoder g~! and a decoder g. The encoder maps the original image yw ⬠R⢠onto its low-dimensional representation z; = goâ (ys) eRâ, where m < M; the decoder maps this feature back to a high- dimensional representation 7 = g(Z). The gray color represents high-dimensional observations.\nHigh-dim. observations Features Control inputs\nFigure 3. Prediction model: Each feature z; is computed from high-dimensional data y; via the encoder g~'. The transition model predicts the feature 2,41)/,,, at the next time step based on the n-step history of n past features z;-n41,..., Z¢ and con- trol inputs weân+1,-.., ut. The predicted feature 241), can be mapped to a high-dimensional prediction #41 via the decoder g. The gray color represents high-dimensional observations.\n# 2.1. Deep Auto-Encoder",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 10 | # 2.1. Deep Auto-Encoder
The learning agent faces the following additional chal- lenges: (a) The agent has no access to the true state, but perceives the environment only through high-dimensional pixel information (images), (b) a good control policy is re- quired in only a few trials. This setting is practically rel- evant, e.g., when the agent is a robot that is monitored by a video camera based on which the robot has to learn to solve tasks fully autonomously. Therefore, this setting is an instance of the pixels to torques problem.
# 2. Deep Dynamical Model | 1502.02251#10 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 10,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 2.1. Deep Auto-Encoder\nThe learning agent faces the following additional chal- lenges: (a) The agent has no access to the true state, but perceives the environment only through high-dimensional pixel information (images), (b) a good control policy is re- quired in only a few trials. This setting is practically rel- evant, e.g., when the agent is a robot that is monitored by a video camera based on which the robot has to learn to solve tasks fully autonomously. Therefore, this setting is an instance of the pixels to torques problem.\n# 2. Deep Dynamical Model",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 11 | # 2. Deep Dynamical Model
We use a deep auto-encoder for embedding images in a low-dimensional feature space, where both the encoder g~! and the decoder g are modeled with deep neural networks. Each layer k of the encoder neural network g~! computes yt) = (Any + by), where o is a sigmoidal acti- vation function (we used arctan) and A, and by are free parameters. The input to the first layer is the image, i.e., (1) Y= Yt The last layer is the low-dimensional fea- ture representation of the image z:(Oz) = g~'(yt;@e), where 6 = [..., Ax, bx, -..] are the parameters of all neu- ral network layers. The decoder g consists of the same number of layers in reverse order, see Fig. 2, and ap- proximately inverts the encoder g, such that %; (9g, 0p) = 9(g~* (yt; 9E); OD) © ys is the reconstructed version of yz with an associated reconstruction error | 1502.02251#11 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 11,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 2. Deep Dynamical Model\nWe use a deep auto-encoder for embedding images in a low-dimensional feature space, where both the encoder g~! and the decoder g are modeled with deep neural networks. Each layer k of the encoder neural network g~! computes yt) = (Any + by), where o is a sigmoidal acti- vation function (we used arctan) and A, and by are free parameters. The input to the first layer is the image, i.e., (1) Y= Yt The last layer is the low-dimensional fea- ture representation of the image z:(Oz) = g~'(yt;@e), where 6 = [..., Ax, bx, -..] are the parameters of all neu- ral network layers. The decoder g consists of the same number of layers in reverse order, see Fig. 2, and ap- proximately inverts the encoder g, such that %; (9g, 0p) = 9(g~* (yt; 9E); OD) © ys is the reconstructed version of yz with an associated reconstruction error",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 12 | Our approach to solve the pixels-to-torques problem is based on a deep dynamical model (DDM), which jointly (i) embeds high-dimensional images in a low-dimensional feature space via deep auto-encoders and (ii) learns a pre- dictive forward model in this feature space (Wahlstro6m et al., 2015). In particular, we consider a DDM with con- trol inputs u and high-dimensional observations y. We as- sume that the relevant properties of y can be compactly represented by a feature variable z. The two components of the DDM, i.e., the low-dimensional embedding and the prediction model, which predicts future observations yt+1 based on past observations and control inputs, are de- tailed in the following. Throughout this paper, y, denotes the high-dimensional measurements, z, the corresponding low-dimensional encoded features and %; the reconstructed high-dimensional measurement. Further, 2,41 and #41 de- note a predicted feature and measurement at time t + 1, respectively.
εR t (θE, θD) = yt (1)
# â Ge(Oe, OD).
The main purpose of the deep auto-encoder is to keep this reconstruction error and the associated compression loss negligible, such that the features zt are a compact repre- sentation of the images yt. | 1502.02251#12 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 12,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Our approach to solve the pixels-to-torques problem is based on a deep dynamical model (DDM), which jointly (i) embeds high-dimensional images in a low-dimensional feature space via deep auto-encoders and (ii) learns a pre- dictive forward model in this feature space (Wahlstro6m et al., 2015). In particular, we consider a DDM with con- trol inputs u and high-dimensional observations y. We as- sume that the relevant properties of y can be compactly represented by a feature variable z. The two components of the DDM, i.e., the low-dimensional embedding and the prediction model, which predicts future observations yt+1 based on past observations and control inputs, are de- tailed in the following. Throughout this paper, y, denotes the high-dimensional measurements, z, the corresponding low-dimensional encoded features and %; the reconstructed high-dimensional measurement. Further, 2,41 and #41 de- note a predicted feature and measurement at time t + 1, respectively.\nεR t (θE, θD) = yt (1)\n# â Ge(Oe, OD).\nThe main purpose of the deep auto-encoder is to keep this reconstruction error and the associated compression loss negligible, such that the features zt are a compact repre- sentation of the images yt.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 13 | # 2.2. Prediction Model
We now turn the static auto-encoder into a dynamical model that can predict future features 2, and images Ji41. The encoder g~+ allows us to map high-dimensional observations y; onto low-dimensional features z;. For pre- dicting we assume that future features 241
,, depend on an n-step history h, of past features and control inputs, ie.,
Zr ajh, (Op) = f (Zt, Ue, ++ Ztâng1, Weng; Op),
(2)
From Pixels to Torques: Policy Learning with Deep Dynamical Models
where f is a nonlinear transition function, in our case a feed-forward neural network, and θP are the correspond- ing model parameters. This is a nonlinear autoregressive exogenous model (NARX) (Ljung, 1999). The predictive performance of the model will be important for model pre- dictive control (see Section 3) and for model learning based on the prediction error (Ljung, 1999).
To predict future observations Y141
,, We exploit the de- coder, such that +1)n, = 9(2:41)n,39D)- The deep de- coder g maps features z to high-dimensional observations y parameterized by Op. | 1502.02251#13 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 13,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 2.2. Prediction Model\nWe now turn the static auto-encoder into a dynamical model that can predict future features 2, and images Ji41. The encoder g~+ allows us to map high-dimensional observations y; onto low-dimensional features z;. For pre- dicting we assume that future features 241\n,, depend on an n-step history h, of past features and control inputs, ie.,\nZr ajh, (Op) = f (Zt, Ue, ++ Ztâng1, Weng; Op),\n(2)\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\nwhere f is a nonlinear transition function, in our case a feed-forward neural network, and θP are the correspond- ing model parameters. This is a nonlinear autoregressive exogenous model (NARX) (Ljung, 1999). The predictive performance of the model will be important for model pre- dictive control (see Section 3) and for model learning based on the prediction error (Ljung, 1999).\nTo predict future observations Y141\n,, We exploit the de- coder, such that +1)n, = 9(2:41)n,39D)- The deep de- coder g maps features z to high-dimensional observations y parameterized by Op.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 14 | cost function is minimized by the BFGS algorithm (No- cedal & Wright, 2006). Note that in (5a) it is crucial to include not only the prediction error VP, but also the re- construction error VR. Without this term the multi-step ahead prediction performance will decrease because pre- dicted features are not consistent with features achieved from the encoder. Since we consider a control problem in this paper, multi-step ahead predictive performance is cru- cial.
Now, we are ready to put the pieces together: With feature prediction model (2) and the deep auto-encoder, the DDM predicts future features and images according to
zt(θE) = gâ1(yt; θE),
(3a) n+1; θP), (3b)
Zr4ajn, (Op, Op) = f (Zt; Wes +s Zn gas Urn $13 OP) Tesrjn,, (Oe, Od, OP) = g(Zr41]h,3 9D), (3b) which is illustrated in Fig. 3. With this prediction model we define the prediction error
â
εP t+1(θE, θD, θP) = yt+1 (4)
â Tern, (Ox, 4%, Op), | 1502.02251#14 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 14,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "cost function is minimized by the BFGS algorithm (No- cedal & Wright, 2006). Note that in (5a) it is crucial to include not only the prediction error VP, but also the re- construction error VR. Without this term the multi-step ahead prediction performance will decrease because pre- dicted features are not consistent with features achieved from the encoder. Since we consider a control problem in this paper, multi-step ahead predictive performance is cru- cial.\nNow, we are ready to put the pieces together: With feature prediction model (2) and the deep auto-encoder, the DDM predicts future features and images according to\nzt(θE) = gâ1(yt; θE),\n(3a) n+1; θP), (3b)\nZr4ajn, (Op, Op) = f (Zt; Wes +s Zn gas Urn $13 OP) Tesrjn,, (Oe, Od, OP) = g(Zr41]h,3 9D), (3b) which is illustrated in Fig. 3. With this prediction model we define the prediction error\nâ\nεP t+1(θE, θD, θP) = yt+1 (4)\nâ Tern, (Ox, 4%, Op),",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 15 | â
εP t+1(θE, θD, θP) = yt+1 (4)
â Tern, (Ox, 4%, Op),
Initialization. With a linear activation function the auto- encoder and PCA are identical (Bourlard & Kamp, 1988), which we exploit to initialize the parameters of the auto- encoder: The auto-encoder network is unfolded, each pair of layers in the encoder and the decoder are combined, and the corresponding PCA solution is computed for each of these pairs. We start with high-dimensional image data at the top layer and use the principal components from that pair of layers as input to the next pair of layers. Thereby, we recursively compute a good initialization for all parameters of the auto-encoder. Similar pre-training routines are found in (Hinton & Salakhutdinov, 2006), in which a restricted Boltzmann machine is used instead of PCA.
where yt+1 is the observed image at time t + 1.
# 2.3. Training
The DDM is parameterized by the encoder parameters 6p, the decoder parameters @p and the prediction model param- eters Op. In the DDM, we train both the prediction model and the deep auto-encoder jointly by finding parameters (6, , 6p). such that
such that Op) =arg min 8.00 N | 1502.02251#15 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 15,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "â\nεP t+1(θE, θD, θP) = yt+1 (4)\nâ Tern, (Ox, 4%, Op),\nInitialization. With a linear activation function the auto- encoder and PCA are identical (Bourlard & Kamp, 1988), which we exploit to initialize the parameters of the auto- encoder: The auto-encoder network is unfolded, each pair of layers in the encoder and the decoder are combined, and the corresponding PCA solution is computed for each of these pairs. We start with high-dimensional image data at the top layer and use the principal components from that pair of layers as input to the next pair of layers. Thereby, we recursively compute a good initialization for all parameters of the auto-encoder. Similar pre-training routines are found in (Hinton & Salakhutdinov, 2006), in which a restricted Boltzmann machine is used instead of PCA.\nwhere yt+1 is the observed image at time t + 1.\n# 2.3. Training\nThe DDM is parameterized by the encoder parameters 6p, the decoder parameters @p and the prediction model param- eters Op. In the DDM, we train both the prediction model and the deep auto-encoder jointly by finding parameters (6, , 6p). such that\nsuch that Op) =arg min 8.00 N",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 16 | such that Op) =arg min 8.00 N
(6x, 8p, Op) =arg min Va(9g, Op) + Vo(Oz, Op, Op), (Sa) 8.00
N c Vez, 8; 0) = D>, _, ler Oz, 6, 8). (5b)
N Va(G, 40) = D2, llet @z, 0) |, (Se)
which minimizes the sums of squared reconstruction (1) and prediction (4) errors.
We learn all model parameters θE, θD, θP jointly by solv- ing (5a).1 The required gradients with respect to the param- eters are computed efï¬ciently by back-propagation, and the
In this section, we have presented a DDM that facili- tates fast predictions of high-dimensional observations via a low-dimensional embedded time series. The property of fast predictions will be exploited by the online feedback control strategy presented in the following. More details on the proposed model are given in (Wahlstr¨om et al., 2015).
# 3. Learning Closed-Loop Policies from Images | 1502.02251#16 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 16,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "such that Op) =arg min 8.00 N\n(6x, 8p, Op) =arg min Va(9g, Op) + Vo(Oz, Op, Op), (Sa) 8.00\nN c Vez, 8; 0) = D>, _, ler Oz, 6, 8). (5b)\nN Va(G, 40) = D2, llet @z, 0) |, (Se)\nwhich minimizes the sums of squared reconstruction (1) and prediction (4) errors.\nWe learn all model parameters θE, θD, θP jointly by solv- ing (5a).1 The required gradients with respect to the param- eters are computed efï¬ciently by back-propagation, and the\nIn this section, we have presented a DDM that facili- tates fast predictions of high-dimensional observations via a low-dimensional embedded time series. The property of fast predictions will be exploited by the online feedback control strategy presented in the following. More details on the proposed model are given in (Wahlstr¨om et al., 2015).\n# 3. Learning Closed-Loop Policies from Images",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 17 | # 3. Learning Closed-Loop Policies from Images
We use the DDM for learning a closed-loop policy by means of nonlinear model predictive control (MPC). We start off by an introduction to classical MPC, before mov- ing on to MPC on images in Section 3.1. MPC ï¬nds an op- timal sequence of control signals that minimizes a K-step loss function, where K is typically smaller than the full horizon. In general, MPC relies on (a) a reference trajec- 1, . . . , xâ tory xref = xâ K (which can be a constant reference signal) and (b) a dynamics model
âNormally when features are used for learning dynamical models, they are first extracted from the data in a pre-processing step by minimizing (5c) with respect to the auto-encoder param- eters 02,4. In a second step, the prediction model parameters Op are estimated based on these features by minimizing (5b) con- ditioned on the estimated 05 and . In our experience, a prob- lem with this approach is that the learned features might have a small reconstruction error, but this representation will not be ideal for learning a transition model. The supplementary material dis- cusses this in more detail.
xt+1 = f (xt, ut), (6) | 1502.02251#17 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 17,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 3. Learning Closed-Loop Policies from Images\nWe use the DDM for learning a closed-loop policy by means of nonlinear model predictive control (MPC). We start off by an introduction to classical MPC, before mov- ing on to MPC on images in Section 3.1. MPC ï¬nds an op- timal sequence of control signals that minimizes a K-step loss function, where K is typically smaller than the full horizon. In general, MPC relies on (a) a reference trajec- 1, . . . , xâ tory xref = xâ K (which can be a constant reference signal) and (b) a dynamics model\nâNormally when features are used for learning dynamical models, they are first extracted from the data in a pre-processing step by minimizing (5c) with respect to the auto-encoder param- eters 02,4. In a second step, the prediction model parameters Op are estimated based on these features by minimizing (5b) con- ditioned on the estimated 05 and . In our experience, a prob- lem with this approach is that the learned features might have a small reconstruction error, but this representation will not be ideal for learning a transition model. The supplementary material dis- cusses this in more detail.\nxt+1 = f (xt, ut), (6)",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 18 | xt+1 = f (xt, ut), (6)
which, assuming that the current state is denoted by xo, can be used to compute/predict a state trajectory Z1,...,£« for a given sequence uo,...,WuxKâ1 of control signals. Using the dynamics model MPC determines an optimal (open- loop) control sequence ug,...,Uj,_,, such that the pre- dicted trajectory %1,...,2« gets as close to the reference
From Pixels to Torques: Policy Learning with Deep Dynamical Models
trajectory xref as possible, such that
K-1 Up,.+., Uz, ⬠arg min > |Z, â a |? + Aljuel|?, Uu0o:K-1 i=0
where ||7; â 27||? is a cost associated with the deviation of the predicted state trajectory Zo. ; from the reference tra- jectory ayer, and ||u;||? penalizes the amplitude of the con- trol signals. Note that the predicted £, depends on all pre- vious ug:7â1. When the control sequence up,...,Uj_1 is determined, the first control ug is applied to the system. After observing the next state, MPC repeats the entire op- timization and turns the overall policy into a closed-loop (feedback) control strategy. | 1502.02251#18 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 18,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "xt+1 = f (xt, ut), (6)\nwhich, assuming that the current state is denoted by xo, can be used to compute/predict a state trajectory Z1,...,£« for a given sequence uo,...,WuxKâ1 of control signals. Using the dynamics model MPC determines an optimal (open- loop) control sequence ug,...,Uj,_,, such that the pre- dicted trajectory %1,...,2« gets as close to the reference\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\ntrajectory xref as possible, such that\nK-1 Up,.+., Uz, ⬠arg min > |Z, â a |? + Aljuel|?, Uu0o:K-1 i=0\nwhere ||7; â 27||? is a cost associated with the deviation of the predicted state trajectory Zo. ; from the reference tra- jectory ayer, and ||u;||? penalizes the amplitude of the con- trol signals. Note that the predicted £, depends on all pre- vious ug:7â1. When the control sequence up,...,Uj_1 is determined, the first control ug is applied to the system. After observing the next state, MPC repeats the entire op- timization and turns the overall policy into a closed-loop (feedback) control strategy.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 19 | # 3.1. MPC on Images
of our MPC formulation lies the DDM, which is used to predict future states (8) from a sequence of control inputs. The quality of the MPC controller is inherently bound to the prediction quality of the dynamical model, which is typical in model-based RL (Schneider, 1997; Schaal, 1997; Deisenroth et al., 2015).
To learn models and controllers from scratch, we apply a control scheme that allows us to update the DDM as new data arrives. In particular, we use the MPC controller in an adaptive fashion to gradually improve the model by col- lected data in the feedback loop without any speciï¬c prior knowledge of the system at hand. Data collection is per- formed in closed-loop (online MPC), and it is divided into multiple sequential trials. After each trial, we add the data of the most recent trajectory to the data set, and the model is re-trained using all data that has been collected so far. | 1502.02251#19 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 19,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 3.1. MPC on Images\nof our MPC formulation lies the DDM, which is used to predict future states (8) from a sequence of control inputs. The quality of the MPC controller is inherently bound to the prediction quality of the dynamical model, which is typical in model-based RL (Schneider, 1997; Schaal, 1997; Deisenroth et al., 2015).\nTo learn models and controllers from scratch, we apply a control scheme that allows us to update the DDM as new data arrives. In particular, we use the MPC controller in an adaptive fashion to gradually improve the model by col- lected data in the feedback loop without any speciï¬c prior knowledge of the system at hand. Data collection is per- formed in closed-loop (online MPC), and it is divided into multiple sequential trials. After each trial, we add the data of the most recent trajectory to the data set, and the model is re-trained using all data that has been collected so far.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 20 | We now turn the classical MPC procedure into MPC on im- ages by exploiting some convenient properties of the DDM. The DDM allows us to predict features 21,...,2« based on a sequence of controls uo, ..., ux â1. By comparing (6) with (2), we define the state xo as the present and past nâ1 features and the past n â 1 control inputs, such that
â
x0 = [z0, . . . , zân+1, uâ1, . . . , uân+1]. (8)
The DDM computes the present and past features with the encoder zt = gâ1(yt, θE), such that x0 is known at the current time, which matches the MPC requirement. Our objective is to control the system towards a desired refer- ence image frame yref. This reference frame yref can also be encoded to a corresponding reference feature zref = gâ1(yref, θE), which results in the MPC objective
K-1 Up,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9 | 1502.02251#20 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 20,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "We now turn the classical MPC procedure into MPC on im- ages by exploiting some convenient properties of the DDM. The DDM allows us to predict features 21,...,2« based on a sequence of controls uo, ..., ux â1. By comparing (6) with (2), we define the state xo as the present and past nâ1 features and the past n â 1 control inputs, such that\nâ\nx0 = [z0, . . . , zân+1, uâ1, . . . , uân+1]. (8)\nThe DDM computes the present and past features with the encoder zt = gâ1(yt, θE), such that x0 is known at the current time, which matches the MPC requirement. Our objective is to control the system towards a desired refer- ence image frame yref. This reference frame yref can also be encoded to a corresponding reference feature zref = gâ1(yref, θE), which results in the MPC objective\nK-1 Up,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 21 | K-1 Up,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9
Up,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9 where x, defined in (8), is the current state. The gradi- ents of the cost function (9) with respect to the control sig- nals uo,...,WKâ1 are computed in closed form, and we use BFGS to find the optimal sequence of control signals. Note that the objective function depends on uo,...,uKâ1 not only via the control penalty |||? but also via the fea- ture predictions 21.â1 of the DDM via (2). Overall, we now have an online MPC algorithm that, given a trained DDM, works indirectly on images by exploiting their feature representation. In the following, we will now turn this into an iterative algorithm that learns predictive models from images and good controllers from scratch.
Algorithm 1 Adaptive MPC in feature space | 1502.02251#21 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 21,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "K-1 Up,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9\nUp,-++,Uxâ1 © arg min > 2: â zreel|? +Alfuell?, (9) uoK-1 4=9 where x, defined in (8), is the current state. The gradi- ents of the cost function (9) with respect to the control sig- nals uo,...,WKâ1 are computed in closed form, and we use BFGS to find the optimal sequence of control signals. Note that the objective function depends on uo,...,uKâ1 not only via the control penalty |||? but also via the fea- ture predictions 21.â1 of the DDM via (2). Overall, we now have an online MPC algorithm that, given a trained DDM, works indirectly on images by exploiting their feature representation. In the following, we will now turn this into an iterative algorithm that learns predictive models from images and good controllers from scratch.\nAlgorithm 1 Adaptive MPC in feature space",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 22 | Algorithm 1 Adaptive MPC in feature space
Algorithm 1 Adaptive MPC in feature space Follow a random control strategy and record data loop Update DDM with all data collected so far for t = 0 to Nâ1do Get state x; via auto-encoder uy < â¬-greedy MPC policy using DDM prediction Apply uj and record data end for end loop
Simply applying the MPC controller based on a randomly initialized model would make the closed-loop system very likely to converge to a point, which is far away from the desired reference value, due to the poor model that can- not extrapolate well to unseen states. This would in turn imply that no data is collected in unexplored regions, in- cluding the region that we actually are interested in. There are two solutions to this problem: Either we use a proba- bilistic dynamics model as suggested in (Schneider, 1997; Deisenroth et al., 2015) to explicitly account for model un- certainty and the implied natural exploration or we follow an explicit exploration strategy to ensure proper excitation of the system. In this paper, we follow the latter approach. In particular, we choose an e-greedy exploration strategy where the optimal feedback uw at each time step is selected with a probability 1 â ¢, and a random action is selected with probability e.
# 3.2. Adaptive MPC for Learning from Scratch | 1502.02251#22 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 22,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Algorithm 1 Adaptive MPC in feature space\nAlgorithm 1 Adaptive MPC in feature space Follow a random control strategy and record data loop Update DDM with all data collected so far for t = 0 to Nâ1do Get state x; via auto-encoder uy < â¬-greedy MPC policy using DDM prediction Apply uj and record data end for end loop\nSimply applying the MPC controller based on a randomly initialized model would make the closed-loop system very likely to converge to a point, which is far away from the desired reference value, due to the poor model that can- not extrapolate well to unseen states. This would in turn imply that no data is collected in unexplored regions, in- cluding the region that we actually are interested in. There are two solutions to this problem: Either we use a proba- bilistic dynamics model as suggested in (Schneider, 1997; Deisenroth et al., 2015) to explicitly account for model un- certainty and the implied natural exploration or we follow an explicit exploration strategy to ensure proper excitation of the system. In this paper, we follow the latter approach. In particular, we choose an e-greedy exploration strategy where the optimal feedback uw at each time step is selected with a probability 1 â ¢, and a random action is selected with probability e.\n# 3.2. Adaptive MPC for Learning from Scratch",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 23 | # 3.2. Adaptive MPC for Learning from Scratch
We will now turn over to describe how (adaptive) MPC can be used together with our DDM to address the pixels to torques problem and to learn from scratch. At the core
Algorithm | summarizes our adaptive online MPC scheme. We initialize the DDM with a random trial. We use the learned DDM to find an e-greedy policy using predicted features within MPC. This happens online. The collected data is added to the data set and the DDM is updated after each trial.
From Pixels to Torques: Policy Learning with Deep Dynamical Models
True video frames Yeo Yer Yer2 Yer3 Vera Yes Yer6 YT Yes Predicted video frames Yerole -Yerrle â-Yerait âYersie â Yerale â Yersie Yer olt t+sie
Figure 4. Long-term (up to eight steps) predictive performance of the DDM: True (upper plot) and predicted (lower plot) video frames on test data. | 1502.02251#23 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 23,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 3.2. Adaptive MPC for Learning from Scratch\nWe will now turn over to describe how (adaptive) MPC can be used together with our DDM to address the pixels to torques problem and to learn from scratch. At the core\nAlgorithm | summarizes our adaptive online MPC scheme. We initialize the DDM with a random trial. We use the learned DDM to find an e-greedy policy using predicted features within MPC. This happens online. The collected data is added to the data set and the DDM is updated after each trial.\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\nTrue video frames Yeo Yer Yer2 Yer3 Vera Yes Yer6 YT Yes Predicted video frames Yerole -Yerrle â-Yerait âYersie â Yerale â Yersie Yer olt t+sie\nFigure 4. Long-term (up to eight steps) predictive performance of the DDM: True (upper plot) and predicted (lower plot) video frames on test data.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 24 | # 4. Experimental Results
(a) Autoencoder and prediction model
In the following, we empirically assess the components of our proposed methodology for autonomous learning from high-dimensional synthetic image data: (a) the quality of the learned DDM and (b) the overall learning framework.
In both cases, we consider a sequence of images (51 51 = 2601 pixels) and a control input associated with these im- ages. Each pixel y(i) is a component of the measurement t R2601 and assumes a continuous gray-value in the in- yt terval [0, 1]. No access to the underlying dynamics or the state (angle Ï and angular velocity ËÏ) was available, i.e., we are dealing with a high-dimensional continuous state space. The challenge was to learn (a) a good dynamics model (b) a good controller from pixel information only. We used a sampling frequency of 0.2 s and a time horizon of 25 s, which corresponds to 100 frames per trial. | 1502.02251#24 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 24,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 4. Experimental Results\n(a) Autoencoder and prediction model\nIn the following, we empirically assess the components of our proposed methodology for autonomous learning from high-dimensional synthetic image data: (a) the quality of the learned DDM and (b) the overall learning framework.\nIn both cases, we consider a sequence of images (51 51 = 2601 pixels) and a control input associated with these im- ages. Each pixel y(i) is a component of the measurement t R2601 and assumes a continuous gray-value in the in- yt terval [0, 1]. No access to the underlying dynamics or the state (angle Ï and angular velocity ËÏ) was available, i.e., we are dealing with a high-dimensional continuous state space. The challenge was to learn (a) a good dynamics model (b) a good controller from pixel information only. We used a sampling frequency of 0.2 s and a time horizon of 25 s, which corresponds to 100 frames per trial.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 25 | The input dimension has been reduced to dim(yt) = 50 prior to model learning using PCA. With these 50- dimensional inputs, a four-layer auto-encoder network was used with dimension 50-25-12-6-2, such that the features were of dimension dim(zt) = 2, which is optimal to model the periodic angle of the pendulum. The order of the dy- namics was selected to be n = 2 (i.e., we consider two consecutive image frames) to capture velocity information, such that zt+1 = f (zt, ut, ztâ1, utâ1). For the prediction model f we used a feedforward neural network with a 6-4- 2 architecture. Note that the dimension of the ï¬rst layer is given by n(dim(zt) + dim(ut)) = 2(2 + 1) = 6.
# (b) Only auto-encoder | 1502.02251#25 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 25,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "The input dimension has been reduced to dim(yt) = 50 prior to model learning using PCA. With these 50- dimensional inputs, a four-layer auto-encoder network was used with dimension 50-25-12-6-2, such that the features were of dimension dim(zt) = 2, which is optimal to model the periodic angle of the pendulum. The order of the dy- namics was selected to be n = 2 (i.e., we consider two consecutive image frames) to capture velocity information, such that zt+1 = f (zt, ut, ztâ1, utâ1). For the prediction model f we used a feedforward neural network with a 6-4- 2 architecture. Note that the dimension of the ï¬rst layer is given by n(dim(zt) + dim(ut)) = 2(2 + 1) = 6.\n# (b) Only auto-encoder",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 26 | # (b) Only auto-encoder
Figure 5. Feature space for both joint (a) and sequential training (b) of auto-encoder and prediction model. The feature space is divided into grid points. For each grid point the decoded high- dimensional image is displayed and the feature values for the training data (red) and validation data (yellow) are overlain. For the joint training the feature values reside on a two-dimensional manifold that corresponds to the two-dimensional position of the tile. For the separate training the feature values are scattered with- out structure.
# 4.1. Learning Predictive Models from Pixels
To assess the predictive performance of the DDM, we took 601 screenshots of a moving tile, see Fig. 4. The control inputs are the (random) increments in position in horizontal and vertical directions.
We evaluate the performance of the learned DDM in terms of long-term predictions, which play a central role in MPC for autonomous learning. Long-term predictions are ob- tained by concatenating multiple 1-step ahead predictions.
The performance of the DDM is illustrated in Fig. 4 on a | 1502.02251#26 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 26,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# (b) Only auto-encoder\nFigure 5. Feature space for both joint (a) and sequential training (b) of auto-encoder and prediction model. The feature space is divided into grid points. For each grid point the decoded high- dimensional image is displayed and the feature values for the training data (red) and validation data (yellow) are overlain. For the joint training the feature values reside on a two-dimensional manifold that corresponds to the two-dimensional position of the tile. For the separate training the feature values are scattered with- out structure.\n# 4.1. Learning Predictive Models from Pixels\nTo assess the predictive performance of the DDM, we took 601 screenshots of a moving tile, see Fig. 4. The control inputs are the (random) increments in position in horizontal and vertical directions.\nWe evaluate the performance of the learned DDM in terms of long-term predictions, which play a central role in MPC for autonomous learning. Long-term predictions are ob- tained by concatenating multiple 1-step ahead predictions.\nThe performance of the DDM is illustrated in Fig. 4 on a",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 27 | The performance of the DDM is illustrated in Fig. 4 on a
test data set. The top row shows the ground truth images and the bottom row shows the DDMâs long-term predic- tions. The model predicts future frames of the tile with high accuracy both for 1-step ahead and multiple steps ahead. The model yields a good predictive performance for both one-step ahead prediction and multiple-step ahead predic- tion.
In Fig. 5(a), the feature representation of the data is dis- played. The features reside on a two-dimensional manifold that encodes the two-dimensional position of the moving
From Pixels to Torques: Policy Learning with Deep Dynamical Models | 1502.02251#27 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 27,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "The performance of the DDM is illustrated in Fig. 4 on a\ntest data set. The top row shows the ground truth images and the bottom row shows the DDMâs long-term predic- tions. The model predicts future frames of the tile with high accuracy both for 1-step ahead and multiple steps ahead. The model yields a good predictive performance for both one-step ahead prediction and multiple-step ahead predic- tion.\nIn Fig. 5(a), the feature representation of the data is dis- played. The features reside on a two-dimensional manifold that encodes the two-dimensional position of the moving\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 28 | Ist trial 4th trial 7th trial Angle [rad] Angle [rad] Time [s] Time [s] Time [s]
Figure 7. Control performance after 1st to 15th trial evaluated with ε = 0 for 16 different experiments. The objective was to reach an angle of ±Ï.
Figure 6. The feature space z â [â1, 1] Ã [â1, 1] is divided into 9 Ã 9 grid points for illustration purposes. For each grid point the decoded high-dimensional image is displayed. Green: Feature values that correspond to collected experience in previous trials. Cyan: Feature value that corresponds to the current time step. Red: Desired reference value. Yellow: 15-steps-ahead prediction after optimizing for the optimal control inputs.
tile. By inspecting the decoded images we can see that each corner of the manifold corresponds to a corner po- sition of the tile. Due to this structure a relatively simple prediction model is sufï¬cient to describe the dynamics. In case the auto-encoder and the prediction model would have been learned sequentially (ï¬rst training the auto-encoder, and then based on these features values train the predic- tion model) such a structure would not have been enforced. In Fig. 5(b) the corresponding feature representation is displayed where only the auto-encoder has been trained. Clearly, these features does not exhibit such a structure. | 1502.02251#28 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 28,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Ist trial 4th trial 7th trial Angle [rad] Angle [rad] Time [s] Time [s] Time [s]\nFigure 7. Control performance after 1st to 15th trial evaluated with ε = 0 for 16 different experiments. The objective was to reach an angle of ±Ï.\nFigure 6. The feature space z â [â1, 1] à [â1, 1] is divided into 9 à 9 grid points for illustration purposes. For each grid point the decoded high-dimensional image is displayed. Green: Feature values that correspond to collected experience in previous trials. Cyan: Feature value that corresponds to the current time step. Red: Desired reference value. Yellow: 15-steps-ahead prediction after optimizing for the optimal control inputs.\ntile. By inspecting the decoded images we can see that each corner of the manifold corresponds to a corner po- sition of the tile. Due to this structure a relatively simple prediction model is sufï¬cient to describe the dynamics. In case the auto-encoder and the prediction model would have been learned sequentially (ï¬rst training the auto-encoder, and then based on these features values train the predic- tion model) such a structure would not have been enforced. In Fig. 5(b) the corresponding feature representation is displayed where only the auto-encoder has been trained. Clearly, these features does not exhibit such a structure.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 29 | # 4.2. Closed-Loop Policy Learning from Pixels
the DDM using all collected data so far, where we also in- clude the reference image while learning the auto-encoder.
Fig. 6 displays the decoded images corresponding to 1, 1]2. The learned fea- learned latent representations in [ ture values of the training data (green) line up in a circular shape, such that a relatively simple prediction model is suf- ï¬cient to describe the dynamics. If we would not have opti- mized for both the prediction error and reconstruction error, such an advantageous structure of the feature values would not have been obtained. The DDM extracts features that can also model the dynamic behavior compactly. The ï¬gure also shows the predictions produced by the MPC controller (yellow), starting from the current time step (cyan) and tar- geting the reference feature (red) where the pendulum is in the target position.
To assess the controller performance after each trial, we applied a greedy policy (⬠= 0). In Fig. 7, angle trajectories for 15 of the 50 experiments at different learning stages are displayed. In the first trial, the controller managed only ina few cases to drive the pendulum toward the reference value +t. The control performance increased gradually with the number of trials, and after the 15th trial, it manages in most cases to get it to an upright position. | 1502.02251#29 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 29,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 4.2. Closed-Loop Policy Learning from Pixels\nthe DDM using all collected data so far, where we also in- clude the reference image while learning the auto-encoder.\nFig. 6 displays the decoded images corresponding to 1, 1]2. The learned fea- learned latent representations in [ ture values of the training data (green) line up in a circular shape, such that a relatively simple prediction model is suf- ï¬cient to describe the dynamics. If we would not have opti- mized for both the prediction error and reconstruction error, such an advantageous structure of the feature values would not have been obtained. The DDM extracts features that can also model the dynamic behavior compactly. The ï¬gure also shows the predictions produced by the MPC controller (yellow), starting from the current time step (cyan) and tar- geting the reference feature (red) where the pendulum is in the target position.\nTo assess the controller performance after each trial, we applied a greedy policy (⬠= 0). In Fig. 7, angle trajectories for 15 of the 50 experiments at different learning stages are displayed. In the first trial, the controller managed only ina few cases to drive the pendulum toward the reference value +t. The control performance increased gradually with the number of trials, and after the 15th trial, it manages in most cases to get it to an upright position.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 30 | In this section, we report results on learning a policy that moves a pendulum (1-link robot arm with length 1m, weight | kg and friction coefficient 1 Nsm/rad) from a start position y = 0 to a target position y = +7. The reference signal was the screenshot of the pendulum in the target po- sition. For the MPC controller, we used a planning horizon of P = 15 steps and a control penalty \ = 0.01. For the e-greedy exploration strategy we used ⬠= 0.2. We con- ducted 50 independent experiments with different random initializations. The learning algorithm was run for 15 trials (plus an initial random trial). After each trial, we retrained | 1502.02251#30 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 30,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "In this section, we report results on learning a policy that moves a pendulum (1-link robot arm with length 1m, weight | kg and friction coefficient 1 Nsm/rad) from a start position y = 0 to a target position y = +7. The reference signal was the screenshot of the pendulum in the target po- sition. For the MPC controller, we used a planning horizon of P = 15 steps and a control penalty \\ = 0.01. For the e-greedy exploration strategy we used ⬠= 0.2. We con- ducted 50 independent experiments with different random initializations. The learning algorithm was run for 15 trials (plus an initial random trial). After each trial, we retrained",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 31 | To assess the data efï¬ciency of our approach, we compared it with the PILCO RL framework (Deisenroth et al., 2015) to learning closed-loop control policies for the pendulum task above. PILCO is a current state-of-the art model-based RL algorithm for data-efï¬cient learning of control policies in continuous state-control spaces. Using collected data PILCO learns a probabilistic model of the system dynam- ics, implemented as a Gaussian process (GP) (Rasmussen & Williams, 2006). Subsequently, this model is used to compute a distribution over trajectories and the correspondFrom Pixels to Torques: Policy Learning with Deep Dynamical Models
1 0.8 e t a R s s e c c u S 0.6 0.4 0.2 PILCO w/ 2D state (Ï, ËÏ) PILCO w/ 2D AE features PILCO w/ 20D PCA features DDM+MPC 0 0 500 1,000 1,500
separately. The auto-encoder ï¬nds good features that min- imize the reconstruction error. However, these features are not good for modeling the dynamic behavior of the sys- tem,3 and lead to bad long-term predictions. | 1502.02251#31 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 31,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "To assess the data efï¬ciency of our approach, we compared it with the PILCO RL framework (Deisenroth et al., 2015) to learning closed-loop control policies for the pendulum task above. PILCO is a current state-of-the art model-based RL algorithm for data-efï¬cient learning of control policies in continuous state-control spaces. Using collected data PILCO learns a probabilistic model of the system dynam- ics, implemented as a Gaussian process (GP) (Rasmussen & Williams, 2006). Subsequently, this model is used to compute a distribution over trajectories and the correspondFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\n1 0.8 e t a R s s e c c u S 0.6 0.4 0.2 PILCO w/ 2D state (Ï, ËÏ) PILCO w/ 2D AE features PILCO w/ 20D PCA features DDM+MPC 0 0 500 1,000 1,500 \nseparately. The auto-encoder ï¬nds good features that min- imize the reconstruction error. However, these features are not good for modeling the dynamic behavior of the sys- tem,3 and lead to bad long-term predictions.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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1502.02251 | 32 | Computation times of PILCO and our method are vastly different: While PILCO spends most time optimizing pol- icy parameters, our model spends most of the time on learn- ing the DDM. Computing the optimal nonparametric MPC policy happens online and does not require signiï¬cant com- putational overhead. To put this into context, PILCO re- quired a few days of learning time for 10 trials (in a 20D feature space). In a 2D feature space, running PILCO for 10 trials and 1000 data points requires about 10 hours.
# Number of frames (100 per trial)
Figure 8. Average learning success with standard errors. Blue: PILCO ground-truth RL baseline using the true state (Ï, ËÏ). Red: PILCO with learned auto-encoder features from image pixels. Cyan: PILCO on 20D feature determined by PCA. Black: Our proposed MPC solution using the DDM.
ing expected cost, which is used for gradient-based opti- mization of the controller parameters. | 1502.02251#32 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 32,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Computation times of PILCO and our method are vastly different: While PILCO spends most time optimizing pol- icy parameters, our model spends most of the time on learn- ing the DDM. Computing the optimal nonparametric MPC policy happens online and does not require signiï¬cant com- putational overhead. To put this into context, PILCO re- quired a few days of learning time for 10 trials (in a 20D feature space). In a 2D feature space, running PILCO for 10 trials and 1000 data points requires about 10 hours.\n# Number of frames (100 per trial)\nFigure 8. Average learning success with standard errors. Blue: PILCO ground-truth RL baseline using the true state (Ï, ËÏ). Red: PILCO with learned auto-encoder features from image pixels. Cyan: PILCO on 20D feature determined by PCA. Black: Our proposed MPC solution using the DDM.\ning expected cost, which is used for gradient-based opti- mization of the controller parameters.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 33 | ing expected cost, which is used for gradient-based opti- mization of the controller parameters.
Although PILCO uses data very efï¬ciently, its computa- tional demand makes its direct application impractical for 20 D) problems, many data points or high-dimensional ( such that we had to make suitable adjustments to apply PILCO to the pixels-to-torques problem. In particular, we performed the following experiments: (1) PILCO applied to 20D PCA features, (2) PILCO applied to 2D features learned by deep auto-encoders, (3) An optimal baseline where we applied PILCO to the standard RL setting with access to the âtrueâ state (Ï, ËÏ) (Deisenroth et al., 2015).
Overall, our DDM+MPC approach to learning closed-loop policies from high-dimensional observations exploits the learned Deep Dynamical Model to learn good policies fairly data efï¬ciently.
# 5. Conclusion | 1502.02251#33 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 33,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "ing expected cost, which is used for gradient-based opti- mization of the controller parameters.\nAlthough PILCO uses data very efï¬ciently, its computa- tional demand makes its direct application impractical for 20 D) problems, many data points or high-dimensional ( such that we had to make suitable adjustments to apply PILCO to the pixels-to-torques problem. In particular, we performed the following experiments: (1) PILCO applied to 20D PCA features, (2) PILCO applied to 2D features learned by deep auto-encoders, (3) An optimal baseline where we applied PILCO to the standard RL setting with access to the âtrueâ state (Ï, ËÏ) (Deisenroth et al., 2015).\nOverall, our DDM+MPC approach to learning closed-loop policies from high-dimensional observations exploits the learned Deep Dynamical Model to learn good policies fairly data efï¬ciently.\n# 5. Conclusion",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 34 | # 5. Conclusion
We have proposed a data-efï¬cient model-based RL algo- rithm that learns closed-loop policies in continuous state and action spaces directly from pixel information. The key components of our solution are (1) a deep dynamical model (DDM) that is used for long-term predictions in a compact feature space and (2) an MPC controller that uses the pre- dictions of the DDM to determine optimal actions on the ï¬y without the need for value function estimation. For the suc- cess of this RL algorithm it is crucial that the DDM learns the feature mapping and the predictive model in feature space jointly to capture dynamic behavior for high-quality long-term predictions. Compared to state-of-the-art RL our algorithm learns fairly quickly, scales to high-dimensional state spaces and facilitates learning from pixels to torques.
# Acknowledgments | 1502.02251#34 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 34,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# 5. Conclusion\nWe have proposed a data-efï¬cient model-based RL algo- rithm that learns closed-loop policies in continuous state and action spaces directly from pixel information. The key components of our solution are (1) a deep dynamical model (DDM) that is used for long-term predictions in a compact feature space and (2) an MPC controller that uses the pre- dictions of the DDM to determine optimal actions on the ï¬y without the need for value function estimation. For the suc- cess of this RL algorithm it is crucial that the DDM learns the feature mapping and the predictive model in feature space jointly to capture dynamic behavior for high-quality long-term predictions. Compared to state-of-the-art RL our algorithm learns fairly quickly, scales to high-dimensional state spaces and facilitates learning from pixels to torques.\n# Acknowledgments",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 35 | # Acknowledgments
Fig. 8 displays the average success rate of PILCO (in- cluding standard error) and our proposed method using deep dynamical models together with a tailored MPC (DDM+MPC). We deï¬ne âsuccessâ if the pendulumâs an- gle is stabilized within 10⦠around the target state.2 The baseline (PILCO trained on the ground-truth 2D state (Ï, ËÏ)) is shown in blue and solves the task very quickly. The graph shows that our proposed algorithm (black), which learns torques directly from pixels, is not too far behind the ground-truth RL solution, achieving a n almost 90% success rate after 15 trials (1500 image frames). How- ever, PILCO trained on the 2D auto-encoder features (red) and 20D PCA features fail consistently in all experiments We explain PILCOâs failure by the fact that we trained the auto-encoder and the transition dynamics in feature space
2Since we consider a continuous setting, we have to deï¬ne a target region. | 1502.02251#35 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 35,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "# Acknowledgments\nFig. 8 displays the average success rate of PILCO (in- cluding standard error) and our proposed method using deep dynamical models together with a tailored MPC (DDM+MPC). We deï¬ne âsuccessâ if the pendulumâs an- gle is stabilized within 10⦠around the target state.2 The baseline (PILCO trained on the ground-truth 2D state (Ï, ËÏ)) is shown in blue and solves the task very quickly. The graph shows that our proposed algorithm (black), which learns torques directly from pixels, is not too far behind the ground-truth RL solution, achieving a n almost 90% success rate after 15 trials (1500 image frames). How- ever, PILCO trained on the 2D auto-encoder features (red) and 20D PCA features fail consistently in all experiments We explain PILCOâs failure by the fact that we trained the auto-encoder and the transition dynamics in feature space\n2Since we consider a continuous setting, we have to deï¬ne a target region.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 36 | 2Since we consider a continuous setting, we have to deï¬ne a target region.
This work was supported by the Swedish Foundation for Strategic Research under the project Cooperative Localiza- tion and the Swedish Research Council under the project Probabilistic modeling of dynamical systems (Contract number: 621-2013-5524). MPD was supported by an Im- perial College Junior Research Fellowship.
# References
Abramova, Ekatarina, Dickens, Luke, Kuhn, Daniel, and Faisal, A. Aldo. Hierarchical, heterogeneous control us- ing reinforcement learning. In EWRL, 2012.
3When we inspected the latent-space embedding of the auto- encoder, the pendulum angles do not nicely line up along an âeasyâ manifold as in Fig. 6. See supplementary material for more details.
From Pixels to Torques: Policy Learning with Deep Dynamical Models
Atkeson, Christopher G. and Schaal, S. Learning tasks from a single demonstration. In ICRA, 1997.
LeCun, Y, Bottou, L, Bengio, Y, and Haffner, P. Gradient- based learning applied to document recognition. Proc. of the IEEE, 86(11):2278â2324, 1998.
Bagnell, James A. and Schneider, Jeff G. Autonomous helicopter control using reinforcement learning policy search methods. In ICRA, 2001. | 1502.02251#36 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 36,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "2Since we consider a continuous setting, we have to deï¬ne a target region.\nThis work was supported by the Swedish Foundation for Strategic Research under the project Cooperative Localiza- tion and the Swedish Research Council under the project Probabilistic modeling of dynamical systems (Contract number: 621-2013-5524). MPD was supported by an Im- perial College Junior Research Fellowship.\n# References\nAbramova, Ekatarina, Dickens, Luke, Kuhn, Daniel, and Faisal, A. Aldo. Hierarchical, heterogeneous control us- ing reinforcement learning. In EWRL, 2012.\n3When we inspected the latent-space embedding of the auto- encoder, the pendulum angles do not nicely line up along an âeasyâ manifold as in Fig. 6. See supplementary material for more details.\nFrom Pixels to Torques: Policy Learning with Deep Dynamical Models\nAtkeson, Christopher G. and Schaal, S. Learning tasks from a single demonstration. In ICRA, 1997.\nLeCun, Y, Bottou, L, Bengio, Y, and Haffner, P. Gradient- based learning applied to document recognition. Proc. of the IEEE, 86(11):2278â2324, 1998.\nBagnell, James A. and Schneider, Jeff G. Autonomous helicopter control using reinforcement learning policy search methods. In ICRA, 2001.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 37 | Bagnell, James A. and Schneider, Jeff G. Autonomous helicopter control using reinforcement learning policy search methods. In ICRA, 2001.
Levine, Sergey, Finn, Chelsea, Darrell, Trevor, and Abbeel, Pieter. End-to-end training of deep visuomotor policies. arXiv preprint arXiv:1504.00702, 2015.
Bengio, Yoshua, Lamblin, Pascal, Popovici, Dan, and Larochelle, Hugo. Greedy layer-wise training of deep networks. In NIPS, 2007.
Ljung, L. System Identiï¬cation: Theory for the User. Pren- tice Hall, 1999.
Boedecker, Joschka, Springenberg, Jost Tobias, W¨ulï¬ng, Jan, and Riedmiller, Martin. Approximate real-time op- timal control based on sparse Gaussian process models. In ADPRL, 2014.
Boots, Byron, Byravan, Arunkumar, and Fox, Dieter. Learning predictive models of a depth camera & manip- ulator from raw execution traces. In ICRA, 2014.
Mayne, David Q. Model predictive control: Recent devel- opments and future promise. Automatica, 50(12):2967â 2986, 2014. | 1502.02251#37 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 37,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Bagnell, James A. and Schneider, Jeff G. Autonomous helicopter control using reinforcement learning policy search methods. In ICRA, 2001.\nLevine, Sergey, Finn, Chelsea, Darrell, Trevor, and Abbeel, Pieter. End-to-end training of deep visuomotor policies. arXiv preprint arXiv:1504.00702, 2015.\nBengio, Yoshua, Lamblin, Pascal, Popovici, Dan, and Larochelle, Hugo. Greedy layer-wise training of deep networks. In NIPS, 2007.\nLjung, L. System Identiï¬cation: Theory for the User. Pren- tice Hall, 1999.\nBoedecker, Joschka, Springenberg, Jost Tobias, W¨ulï¬ng, Jan, and Riedmiller, Martin. Approximate real-time op- timal control based on sparse Gaussian process models. In ADPRL, 2014.\nBoots, Byron, Byravan, Arunkumar, and Fox, Dieter. Learning predictive models of a depth camera & manip- ulator from raw execution traces. In ICRA, 2014.\nMayne, David Q. Model predictive control: Recent devel- opments and future promise. Automatica, 50(12):2967â 2986, 2014.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 38 | Mayne, David Q. Model predictive control: Recent devel- opments and future promise. Automatica, 50(12):2967â 2986, 2014.
Mnih, Volodymyr, Kavukcuoglu, Koray, Silver, David, Rusu, Andrei A, Veness, Joel, Bellemare, Marc G, Graves, Alex, Riedmiller, Martin, Fidjeland, Andreas K, Ostrovski, Georg, and et al. Human-level control Nature, 518 through deep reinforcement (7540):529â533, 2015.
Bourlard, Herv´e and Kamp, Yves. Auto-association by multilayer perceptrons and singular value decomposi- tion. Biological Cybernetics, 59(4-5):291â294, 1988.
Nocedal, J. and Wright, S. J. Numerical Optimization. Springer, 2006.
Brock, Oliver. Berlin Summit on Robotics: Conference Re- port, chapter Is Robotics in Need of a Paradigm Shift?, pp. 1â10. 2011.
Contardo, Gabriella, Denoyer, Ludovic, Artieres, Thierry, and Gallinari, Patrick. Learning states representations in POMDP. arXiv preprint arXiv:1312.6042, 2013. | 1502.02251#38 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 38,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Mayne, David Q. Model predictive control: Recent devel- opments and future promise. Automatica, 50(12):2967â 2986, 2014.\nMnih, Volodymyr, Kavukcuoglu, Koray, Silver, David, Rusu, Andrei A, Veness, Joel, Bellemare, Marc G, Graves, Alex, Riedmiller, Martin, Fidjeland, Andreas K, Ostrovski, Georg, and et al. Human-level control Nature, 518 through deep reinforcement (7540):529â533, 2015.\nBourlard, Herv´e and Kamp, Yves. Auto-association by multilayer perceptrons and singular value decomposi- tion. Biological Cybernetics, 59(4-5):291â294, 1988.\nNocedal, J. and Wright, S. J. Numerical Optimization. Springer, 2006.\nBrock, Oliver. Berlin Summit on Robotics: Conference Re- port, chapter Is Robotics in Need of a Paradigm Shift?, pp. 1â10. 2011.\nContardo, Gabriella, Denoyer, Ludovic, Artieres, Thierry, and Gallinari, Patrick. Learning states representations in POMDP. arXiv preprint arXiv:1312.6042, 2013.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
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] |
1502.02251 | 39 | Cuccu, Giuseppe, Luciw, Matthew, Schmidhuber, J¨urgen, and Gomez, Faustino. Intrinsically motivated neuroevo- lution for vision-based reinforcement learning. In ICDL, 2011.
Pan, Yunpeng and Theodorou, Evangelos. Probabilistic dif- ferential dynamic programming. In NIPS, 2014.
Rasmussen, Carl E. and Williams, Christopher K. I. Gaus- sian Processes for Machine Learning. The MIT Press, 2006.
Schaal, Stefan. Learning from demonstration. In NIPS. 1997.
Schmidhuber, J¨urgen. An on-line algorithm for dynamic reinforcement learning and planning in reactive environ- ments. In IJCNN, 1990.
Deisenroth, Marc P., Rasmussen, Carl E., and Peters, Jan. Gaussian process dynamic programming. Neurocomput- ing, 72(7â9):1508â1524, 2009.
Deisenroth, Marc P., Fox, Dieter, and Rasmussen, Carl E. Gaussian processes for data-efï¬cient learning in robotics and control. IEEE-TPAMI, 37(2):408â423, 2015.
Hinton, G and Salakhutdinov, R. Reducing the dimension- ality of data with neural networks. Science, 313:504â 507, 2006. | 1502.02251#39 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 39,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Cuccu, Giuseppe, Luciw, Matthew, Schmidhuber, J¨urgen, and Gomez, Faustino. Intrinsically motivated neuroevo- lution for vision-based reinforcement learning. In ICDL, 2011.\nPan, Yunpeng and Theodorou, Evangelos. Probabilistic dif- ferential dynamic programming. In NIPS, 2014.\nRasmussen, Carl E. and Williams, Christopher K. I. Gaus- sian Processes for Machine Learning. The MIT Press, 2006.\nSchaal, Stefan. Learning from demonstration. In NIPS. 1997.\nSchmidhuber, J¨urgen. An on-line algorithm for dynamic reinforcement learning and planning in reactive environ- ments. In IJCNN, 1990.\nDeisenroth, Marc P., Rasmussen, Carl E., and Peters, Jan. Gaussian process dynamic programming. Neurocomput- ing, 72(7â9):1508â1524, 2009.\nDeisenroth, Marc P., Fox, Dieter, and Rasmussen, Carl E. Gaussian processes for data-efï¬cient learning in robotics and control. IEEE-TPAMI, 37(2):408â423, 2015.\nHinton, G and Salakhutdinov, R. Reducing the dimension- ality of data with neural networks. Science, 313:504â 507, 2006.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02251 | 40 | Hinton, G and Salakhutdinov, R. Reducing the dimension- ality of data with neural networks. Science, 313:504â 507, 2006.
Koutnik, Jan, Cuccu, Giuseppe, Schmidhuber, J¨urgen, and Gomez, Faustino. Evolving large-scale neural networks In GECCO, for vision-based reinforcement learning. 2013.
Schneider, Jeff G. Exploiting model uncertainty estimates for safe dynamic control learning. In NIPS. 1997.
Sha, Daohang. A new neural networks based adaptive model predictive control for unknown multiple variable non-linear systems. IJAMS, 1(2):146â155, 2008.
Sutton, Richard S. and Barto, Andrew G. Reinforcement Learning: An Introduction. The MIT Press, 1998.
van Hoof, Herke, Peters, Jan, and Neumann, Gerhard. Learning of non-parametric control policies with high- dimensional state features. In AISTATS, 2015.
Vincent, P, Larochelle, H, Bengio, Y, and Manzagol, Pierre-Antoine. Extracting and composing robust fea- tures with denoising autoencoders. In ICML, 2008. | 1502.02251#40 | From Pixels to Torques: Policy Learning with Deep Dynamical Models | Data-efficient learning in continuous state-action spaces using very
high-dimensional observations remains a key challenge in developing fully
autonomous systems. In this paper, we consider one instance of this challenge,
the pixels to torques problem, where an agent must learn a closed-loop control
policy from pixel information only. We introduce a data-efficient, model-based
reinforcement learning algorithm that learns such a closed-loop policy directly
from pixel information. The key ingredient is a deep dynamical model that uses
deep auto-encoders to learn a low-dimensional embedding of images jointly with
a predictive model in this low-dimensional feature space. Joint learning
ensures that not only static but also dynamic properties of the data are
accounted for. This is crucial for long-term predictions, which lie at the core
of the adaptive model predictive control strategy that we use for closed-loop
control. Compared to state-of-the-art reinforcement learning methods for
continuous states and actions, our approach learns quickly, scales to
high-dimensional state spaces and is an important step toward fully autonomous
learning from pixels to torques. | http://arxiv.org/pdf/1502.02251 | Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth | stat.ML, cs.LG, cs.RO, cs.SY | 9 pages | null | stat.ML | 20150208 | 20150618 | [
{
"id": "1504.00702"
}
] | {
"authors": "Niklas Wahlström, Thomas B. Schön, Marc Peter Deisenroth",
"chunk_id": 40,
"doc_id": "1502.02251",
"primary_category": "stat.ML",
"published": 20150208,
"source": "http://arxiv.org/pdf/1502.02251",
"summary": "Data-efficient learning in continuous state-action spaces using very\nhigh-dimensional observations remains a key challenge in developing fully\nautonomous systems. In this paper, we consider one instance of this challenge,\nthe pixels to torques problem, where an agent must learn a closed-loop control\npolicy from pixel information only. We introduce a data-efficient, model-based\nreinforcement learning algorithm that learns such a closed-loop policy directly\nfrom pixel information. The key ingredient is a deep dynamical model that uses\ndeep auto-encoders to learn a low-dimensional embedding of images jointly with\na predictive model in this low-dimensional feature space. Joint learning\nensures that not only static but also dynamic properties of the data are\naccounted for. This is crucial for long-term predictions, which lie at the core\nof the adaptive model predictive control strategy that we use for closed-loop\ncontrol. Compared to state-of-the-art reinforcement learning methods for\ncontinuous states and actions, our approach learns quickly, scales to\nhigh-dimensional state spaces and is an important step toward fully autonomous\nlearning from pixels to torques.",
"text": "Hinton, G and Salakhutdinov, R. Reducing the dimension- ality of data with neural networks. Science, 313:504â 507, 2006.\nKoutnik, Jan, Cuccu, Giuseppe, Schmidhuber, J¨urgen, and Gomez, Faustino. Evolving large-scale neural networks In GECCO, for vision-based reinforcement learning. 2013.\nSchneider, Jeff G. Exploiting model uncertainty estimates for safe dynamic control learning. In NIPS. 1997.\nSha, Daohang. A new neural networks based adaptive model predictive control for unknown multiple variable non-linear systems. IJAMS, 1(2):146â155, 2008.\nSutton, Richard S. and Barto, Andrew G. Reinforcement Learning: An Introduction. The MIT Press, 1998.\nvan Hoof, Herke, Peters, Jan, and Neumann, Gerhard. Learning of non-parametric control policies with high- dimensional state features. In AISTATS, 2015.\nVincent, P, Larochelle, H, Bengio, Y, and Manzagol, Pierre-Antoine. Extracting and composing robust fea- tures with denoising autoencoders. In ICML, 2008.",
"title": "From Pixels to Torques: Policy Learning with Deep Dynamical Models",
"year": 2015
} | [
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] |
1502.02072 | 0 | 5 1 0 2
b e F 6 ] L M . t a t s [
1 v 2 7 0 2 0 . 2 0 5 1 : v i X r a
# Massively Multitask Networks for Drug Discovery
Bharath Ramsundar*,â , ⦠Steven Kearnes*,â Patrick Riley⦠Dale Webster⦠David Konerding⦠Vijay Pandeâ (*Equal contribution, â Stanford University, â¦Google Inc.)
[email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
# Abstract | 1502.02072#0 | Massively Multitask Networks for Drug Discovery | Massively multitask neural architectures provide a learning framework for
drug discovery that synthesizes information from many distinct biological
sources. To train these architectures at scale, we gather large amounts of data
from public sources to create a dataset of nearly 40 million measurements
across more than 200 biological targets. We investigate several aspects of the
multitask framework by performing a series of empirical studies and obtain some
interesting results: (1) massively multitask networks obtain predictive
accuracies significantly better than single-task methods, (2) the predictive
power of multitask networks improves as additional tasks and data are added,
(3) the total amount of data and the total number of tasks both contribute
significantly to multitask improvement, and (4) multitask networks afford
limited transferability to tasks not in the training set. Our results
underscore the need for greater data sharing and further algorithmic innovation
to accelerate the drug discovery process. | http://arxiv.org/pdf/1502.02072 | Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande | stat.ML, cs.LG, cs.NE | Preliminary work. Under review by the International Conference on
Machine Learning (ICML) | null | stat.ML | 20150206 | 20150206 | [] | {
"authors": "Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande",
"chunk_id": 0,
"doc_id": "1502.02072",
"primary_category": "stat.ML",
"published": 20150206,
"source": "http://arxiv.org/pdf/1502.02072",
"summary": "Massively multitask neural architectures provide a learning framework for\ndrug discovery that synthesizes information from many distinct biological\nsources. To train these architectures at scale, we gather large amounts of data\nfrom public sources to create a dataset of nearly 40 million measurements\nacross more than 200 biological targets. We investigate several aspects of the\nmultitask framework by performing a series of empirical studies and obtain some\ninteresting results: (1) massively multitask networks obtain predictive\naccuracies significantly better than single-task methods, (2) the predictive\npower of multitask networks improves as additional tasks and data are added,\n(3) the total amount of data and the total number of tasks both contribute\nsignificantly to multitask improvement, and (4) multitask networks afford\nlimited transferability to tasks not in the training set. Our results\nunderscore the need for greater data sharing and further algorithmic innovation\nto accelerate the drug discovery process.",
"text": "5 1 0 2\nb e F 6 ] L M . t a t s [\n1 v 2 7 0 2 0 . 2 0 5 1 : v i X r a\n# Massively Multitask Networks for Drug Discovery\nBharath Ramsundar*,â , ⦠Steven Kearnes*,â Patrick Riley⦠Dale Webster⦠David Konerding⦠Vijay Pandeâ (*Equal contribution, â Stanford University, â¦Google Inc.)\[email protected] [email protected] [email protected] [email protected] [email protected] [email protected]\n# Abstract",
"title": "Massively Multitask Networks for Drug Discovery",
"year": 2015
} | [
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] |
1502.02072 | 1 | # Abstract
Massively multitask neural architectures provide a learning framework for drug discovery that synthesizes information from many distinct bi- ological sources. To train these architectures at scale, we gather large amounts of data from pub- lic sources to create a dataset of nearly 40 mil- lion measurements across more than 200 bio- logical targets. We investigate several aspects of the multitask framework by performing a se- ries of empirical studies and obtain some in- teresting results: (1) massively multitask net- works obtain predictive accuracies signiï¬cantly better than single-task methods, (2) the pre- dictive power of multitask networks improves as additional tasks and data are added, (3) the total amount of data and the total number of tasks both contribute signiï¬cantly to multitask improvement, and (4) multitask networks afford limited transferability to tasks not in the training set. Our results underscore the need for greater data sharing and further algorithmic innovation to accelerate the drug discovery process. | 1502.02072#1 | Massively Multitask Networks for Drug Discovery | Massively multitask neural architectures provide a learning framework for
drug discovery that synthesizes information from many distinct biological
sources. To train these architectures at scale, we gather large amounts of data
from public sources to create a dataset of nearly 40 million measurements
across more than 200 biological targets. We investigate several aspects of the
multitask framework by performing a series of empirical studies and obtain some
interesting results: (1) massively multitask networks obtain predictive
accuracies significantly better than single-task methods, (2) the predictive
power of multitask networks improves as additional tasks and data are added,
(3) the total amount of data and the total number of tasks both contribute
significantly to multitask improvement, and (4) multitask networks afford
limited transferability to tasks not in the training set. Our results
underscore the need for greater data sharing and further algorithmic innovation
to accelerate the drug discovery process. | http://arxiv.org/pdf/1502.02072 | Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande | stat.ML, cs.LG, cs.NE | Preliminary work. Under review by the International Conference on
Machine Learning (ICML) | null | stat.ML | 20150206 | 20150206 | [] | {
"authors": "Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande",
"chunk_id": 1,
"doc_id": "1502.02072",
"primary_category": "stat.ML",
"published": 20150206,
"source": "http://arxiv.org/pdf/1502.02072",
"summary": "Massively multitask neural architectures provide a learning framework for\ndrug discovery that synthesizes information from many distinct biological\nsources. To train these architectures at scale, we gather large amounts of data\nfrom public sources to create a dataset of nearly 40 million measurements\nacross more than 200 biological targets. We investigate several aspects of the\nmultitask framework by performing a series of empirical studies and obtain some\ninteresting results: (1) massively multitask networks obtain predictive\naccuracies significantly better than single-task methods, (2) the predictive\npower of multitask networks improves as additional tasks and data are added,\n(3) the total amount of data and the total number of tasks both contribute\nsignificantly to multitask improvement, and (4) multitask networks afford\nlimited transferability to tasks not in the training set. Our results\nunderscore the need for greater data sharing and further algorithmic innovation\nto accelerate the drug discovery process.",
"text": "# Abstract\nMassively multitask neural architectures provide a learning framework for drug discovery that synthesizes information from many distinct bi- ological sources. To train these architectures at scale, we gather large amounts of data from pub- lic sources to create a dataset of nearly 40 mil- lion measurements across more than 200 bio- logical targets. We investigate several aspects of the multitask framework by performing a se- ries of empirical studies and obtain some in- teresting results: (1) massively multitask net- works obtain predictive accuracies signiï¬cantly better than single-task methods, (2) the pre- dictive power of multitask networks improves as additional tasks and data are added, (3) the total amount of data and the total number of tasks both contribute signiï¬cantly to multitask improvement, and (4) multitask networks afford limited transferability to tasks not in the training set. Our results underscore the need for greater data sharing and further algorithmic innovation to accelerate the drug discovery process.",
"title": "Massively Multitask Networks for Drug Discovery",
"year": 2015
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] |
1502.02072 | 2 | After a suitable target has been identiï¬ed, the ï¬rst step in the drug discovery process is âhit ï¬nding.â Given some druggable target, pharmaceutical companies will screen millions of drug-like compounds in an effort to ï¬nd a few attractive molecules for further optimization. These screens are often automated via robots, but are expensive to perform. Virtual screening attempts to replace or aug- ment the high-throughput screening process by the use of computational methods (Shoichet, 2004). Machine learn- ing methods have frequently been applied to virtual screen- ing by training supervised classiï¬ers to predict interactions between targets and small molecules.
There are a variety of challenges that must be overcome to achieve effective virtual screening. Low hit rates in experimental screens (often only 1â2% of screened com- pounds are active against a given target) result in im- balanced datasets that require special handling for effec- tive learning. For instance, care must be taken to guard against unrealistic divisions between active and inactive compounds (âartiï¬cial enrichmentâ) and against informa- tion leakage due to strong similarity between active com- pounds (âanalog biasâ) (Rohrer & Baumann, 2009). Fur- thermore, the paucity of experimental data means that over- ï¬tting is a perennial thorn. | 1502.02072#2 | Massively Multitask Networks for Drug Discovery | Massively multitask neural architectures provide a learning framework for
drug discovery that synthesizes information from many distinct biological
sources. To train these architectures at scale, we gather large amounts of data
from public sources to create a dataset of nearly 40 million measurements
across more than 200 biological targets. We investigate several aspects of the
multitask framework by performing a series of empirical studies and obtain some
interesting results: (1) massively multitask networks obtain predictive
accuracies significantly better than single-task methods, (2) the predictive
power of multitask networks improves as additional tasks and data are added,
(3) the total amount of data and the total number of tasks both contribute
significantly to multitask improvement, and (4) multitask networks afford
limited transferability to tasks not in the training set. Our results
underscore the need for greater data sharing and further algorithmic innovation
to accelerate the drug discovery process. | http://arxiv.org/pdf/1502.02072 | Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande | stat.ML, cs.LG, cs.NE | Preliminary work. Under review by the International Conference on
Machine Learning (ICML) | null | stat.ML | 20150206 | 20150206 | [] | {
"authors": "Bharath Ramsundar, Steven Kearnes, Patrick Riley, Dale Webster, David Konerding, Vijay Pande",
"chunk_id": 2,
"doc_id": "1502.02072",
"primary_category": "stat.ML",
"published": 20150206,
"source": "http://arxiv.org/pdf/1502.02072",
"summary": "Massively multitask neural architectures provide a learning framework for\ndrug discovery that synthesizes information from many distinct biological\nsources. To train these architectures at scale, we gather large amounts of data\nfrom public sources to create a dataset of nearly 40 million measurements\nacross more than 200 biological targets. We investigate several aspects of the\nmultitask framework by performing a series of empirical studies and obtain some\ninteresting results: (1) massively multitask networks obtain predictive\naccuracies significantly better than single-task methods, (2) the predictive\npower of multitask networks improves as additional tasks and data are added,\n(3) the total amount of data and the total number of tasks both contribute\nsignificantly to multitask improvement, and (4) multitask networks afford\nlimited transferability to tasks not in the training set. Our results\nunderscore the need for greater data sharing and further algorithmic innovation\nto accelerate the drug discovery process.",
"text": "After a suitable target has been identiï¬ed, the ï¬rst step in the drug discovery process is âhit ï¬nding.â Given some druggable target, pharmaceutical companies will screen millions of drug-like compounds in an effort to ï¬nd a few attractive molecules for further optimization. These screens are often automated via robots, but are expensive to perform. Virtual screening attempts to replace or aug- ment the high-throughput screening process by the use of computational methods (Shoichet, 2004). Machine learn- ing methods have frequently been applied to virtual screen- ing by training supervised classiï¬ers to predict interactions between targets and small molecules.\nThere are a variety of challenges that must be overcome to achieve effective virtual screening. Low hit rates in experimental screens (often only 1â2% of screened com- pounds are active against a given target) result in im- balanced datasets that require special handling for effec- tive learning. For instance, care must be taken to guard against unrealistic divisions between active and inactive compounds (âartiï¬cial enrichmentâ) and against informa- tion leakage due to strong similarity between active com- pounds (âanalog biasâ) (Rohrer & Baumann, 2009). Fur- thermore, the paucity of experimental data means that over- ï¬tting is a perennial thorn.",
"title": "Massively Multitask Networks for Drug Discovery",
"year": 2015
} | [
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