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works.arXiv:2103.10051v1  [cs.LG]  18 Mar 2021DATA-FREE MIXED-PRECISION QUANTIZATION USING NOVEL SENSITIVITY METRIC
Donghyun Lee, Minkyoung Cho, Seungwon Lee, Joonho Song, and Changkyu Choi
Samsung Advanced Institute of Technology, Samsung Electronics, South Korea
ABSTRACT
Post-training quantization is a representative technique for
compressing neural networks, making them smaller and more
efficient for deployment on edge devices. However, an in-
accessible user dataset often makes it difficult to ensure the
quality of the quantized neural network in practice. In addi-
tion, existing approaches may use a single uniform bit-width
across the network, resulting in significant accuracy degrada-
tion at extremely low bit-widths. To utilize multiple bit-width,
sensitivity metric plays a key role in balancing accuracy and
compression. In this paper, we propose a novel sensitivity
metric that considers the effect of quantization error on task
loss and interaction with other layers. Moreover, we develop
labeled data generation methods that are not dependent on a
specific operation of the neural network. Our experiments
show that the proposed metric better represents quantization
sensitivity, and generated data are more feasible to be applied
to mixed-precision quantization.
Index Terms —Deep Learning, Quantization, Data Free
1. INTRODUCTION
In recent years, deep neural networks have simplified and en-
abled applications in numerous domains, especially for vision
tasks [1, 2, 3]. Meanwhile, there is a need to minimize the
memory footprint and reduce the network computation cost to
deploy on edge devices. Significant efforts have been made to
reduce the network size or accelerate inference of the neural
network [4, 5, 6]. Several approaches exist to this problem,
and quantization has been studied as one of the most reliable
solutions. In the quantization process, low-bit representations
of both weights and activations introduce quantization noise,
which results in accuracy degradation. To alleviate the accu-
racy loss, retraining or fine-tuning methods are developed by
exploiting extensive training datasets [7, 8, 9].
However, these methods are not applicable in many real-
world scenarios, where the user dataset is inaccessible due to
confidential or personal issues [10]. It is impossible to train
the network and verify the quality of a quantized neural net-
work without a user dataset. Although post-training quantiza-
tion is a frequently suggested method to address this problem
[11, 12], a small dataset is often required to decide the optimal
Equal contribution
(a) Data generation
 (b) Profiling for quantization
(c) Sensitivity measurement
 (d) Mixed-precision inference
Fig. 1 . Overall process of a post-training mixed-precision
quantization method. (a) Generate dataset from pretrained
network. (b) Post-training quantization using statistics. (c)
Measure the sensitivity of each layer. (d) Quantize to higher
precision for a more sensitive layer.
clipping range of each layer. The credential statistics of lay-
ers are important to ensure the performance of the quantized
network in increasing task loss at ultra-low-bit precision.
Considering that most quantization approaches have
used uniform bit allocation across the network, the mixed-
precision quantization approach takes a step further in pre-
serving the accuracy by lifting the limit of those approaches
[13, 14]. As a necessity, several sensitivity measurement
methods for maximizing the effects of mixed-precision have
also been proposed because it is difficult to determine which
sections of the network are comparatively less susceptible
to quantization [15, 16, 17]. To measure the quantization
robustness of activations and weights, it is necessary to an-
alyze the statistics generated during forward and backward
processes by using a reliable dataset. The prior sensitivity
metrics use the difference between outputs of the original
model and the quantized model when quantizing each layer
separately [10, 15]. However, this approach does not con-
sider the interaction of quantization error and other layers.
It cannot be neglected because a lower bit quantization im-
plies a larger quantization error. Other prior studies require
severe approximations to compute higher-order derivatives
efficiently [16, 18].Fig. 2 . Top-2 confidences of each generated data sample on
ResNet50. Confidence means the output values of the soft-
max layer. For all classes except one (Class 744), the gener-
ated labeled data samples pointed their target classes corre-
sponding to the labels with the highest confidence.
In this work, we provide a straightforward method to com-
pute the layer-wise sensitivity for mixed-precision quantiza-
tion, considering the interaction of quantization error. In addi-
tion, we propose a data generation method, which is effective
in the process of post-training mixed-precision quantization,
as shown in Figure 1. Prior works [10, 19] use statistics of
a particular operation (such as batch normalization), and it is
impotent when the networks that do not have the specific op-
eration explicitly. The proposed synthetic data engineering
approach is independent of network structures. Moreover, it
generates labeled data to verify the quantized network.
The remainder of this paper is organized as follows. Sec-
tion 2 describes the data generation method and proposes a
sensitivity measurement metric. Section 3 provides an exper-
imental demonstration of mixed-precision quantization using
the proposed metric and generated data, and the results are
compared with previous approaches.
2. METHODOLOGY
2.1. Data Generation
For a general data generation method, we seek to avoid any
dependency on a specific operation in a convolutional net-
work. As shown in Figure 1(a), we first forward a noisy image
initialized from a uniform distribution in the range between 0
and 255, inclusive. To produce a set of labeled data, we use
a set of class-relevant vectors/matrices, which means one-hot
vectors per class in the classification task. In the forward pass,
the loss is computed as
x
c= argmin
xcLCE(f(y(xc)); vc) +
y(xc) (1)
wherexcis the trainable input feature, vcis the set of class-
relevant vectors/matrices, called the golden set, y(
)is theneural network, and f(
)is an activation function (i.e., soft-
max) used to compute cross-entropy loss ( LCE) with the
golden set. In addition, we reinforce the loss by maximizing
the activation of output of network to enhance the efficiency
of data generation process by referring the prior works for
model interpretation [20, 21]. The calculated loss is prop-
agated in the backward pass and generates input feature x
c
for each class, c2f1:::NumOfClass g. Finally, we have
crafted synthetic data for each class after several iterations of
forward and backward processes.
Figure 2 demonstrates the reliability of crafting labeled
data for each class by showing the top-2 confidences per class
of synthetic data. All the generated data are used not only to
measure the layer-wise sensitivity of the network but also to
obtain the statistics of activations to improve the quality of the
post-training quantized network.
2.2. Sensitivity Metric
The objective of mixed-precision quantization is to allocate
appropriate bits to each layer to reduce the cost (e.g., bit op-
erations [22]) of neural network models while suppressing the
task loss growth. The sensitivity metric is an important factor
for finding the optimal point between the effects of quanti-
zation error and cost. First, we would like to measure the
effect of quantization error on the loss of a network that has
totalLlayers when weights of ithlayer (1iL),Wi
are quantized through quantization function Qk(
)intok-bit.
Given input data xand quantized neural network ^y, we con-
sider the Euclidean distance between y(x)and^y(x)as theL,
the loss of network output, for sensitivity measurement in-
stead of task loss. We can define the effect of quantization
errorQk(Wi)WiofQk(Wi)onLas follows:

Wi(k) =1
NX
x



@L
@(Qk(Wi)Wi)



(2)
whereNdenotes the total size of the dataset, and 
Wi(k)are
gradients for the quantization error. Wiis not variable in post-
training quantization; thus, we can represent Eq. 2 as weight
gradients of quantized parameters by using the chain rule as
follows:
@L
@Qk(W)
@Qk(W)
@(Qk(W)W)'@L
@Qk(W)(3)
The effects of the activation’s quantization error on Lis
represented as activation gradients of quantized activation by
applying the derivation of the formula shown in the previous
equations. Subsequently, we calculate the expectation for the
effects of the quantized network on Lby using the geometric
mean of 
ofm-bit quantized activation Qm(A)and quan-
tized weights Qk(W), which is formulated asE[jS^yj] =LY
i
Ai(mi)
Wi(ki)1
L
(4)
The gradients of the converged single-precision neural
network are not changed for the same data. To measure
the effect of Qk(Wi)Wion other connected layers, we
observe the gradient perturbations of WandA, which are
caused byQk(Wi). Consequently, we can measure the effect
of quantization error on other layers and loss of the network
together, i.e., sensitivity of quantization by using E[jS^yj]
when we only quantize activations or weights of a layer for
which we would like to measure the sensitivity. Expressing
the single-precision as QFP32(
), we have
E[jSWjj] =LY
i
Ai(FP32)
 Wi(ki)1
L
(5)
s:t: k i=(
ki<32i=j
FP32i6=j
for quantization sensitivity metric for Wj. It is straightfor-
ward by using the information of back-propagation of quan-
tized network and considers the effect of quantization error
on other layers.
3. EXPERIMENTS
In this section, we first demonstrate the effectiveness of the
proposed data generation method in post-training quantiza-
tion. Then, we show that the proposed sensitivity metric rep-
resents the quantization sensitivity of the layer effectively by
using our generated data. Finally, sensitivity value by using
various datasets indicates that the proposed data generation
method is also credible in sensitivity measurement.
To demonstrate our methodology, we use classification
models VGG19, ResNet50, and InceptionV3 on the ImageNet
validation dataset.
3.1. Efficacy of Data Generation Method
We evaluate our method using the generated data to determine
the clipping range of each layer in post-training quantization.
Our experiments exploit a simple symmetric quantization ap-
proach, which uses the maximum value of activation as the
clipping range of each layer. Hence, maximum values are
extremely crucial for confining the dynamic range to utilize
the bit-width efficiently, preventing a severe accuracy drop in
low-bit quantization.
For our proposed method, input images are initialized ac-
cording to uniform distribution, which follows standardiza-
tion or normalization. To generate data, we use Adam op-
timizer to optimize the loss function with a learning rate of
0.04, and we found that =1 works best empirically.Model Dataset Top-1 Top-5
VGG19ImageNet 72.31 90.81
Noise 3.45 10.45
ZeroQ - -
Proposed 71.84 90.53
ResNet50ImageNet 75.89 92.74
Noise 11.28 29.88
ZeroQ 75.47 92.62
Proposed 75.68 92.72
InceptionV3ImageNet 77.14 93.43
Noise 62.49 85.00
ZeroQ 76.85 93.31
Proposed 76.83 93.26
Table 1 . Results of post-training quantization (8-bit quantiza-
tion for activations and weights) using different dataset. Ima-
geNet presents the oracle performance in terms of using train-
ing dataset and others are the results of the data-free methods.
For InceptionV3, input data are initialized according to
U(0, 255), which follows standardization with mean and vari-
ance by considering that the model requires synthetic data to
be converted into the range of [-1, 1] through preprocessing.
For VGG19 and ResNet50, input data are initialized accord-
ing toU(0, 255), which follows normalization with the factor
of 255 because they require the range of [0, 1].
Table 1 shows the empirical results for ImageNet, random
noise, and ZeroQ [10]. In the experiment, ImageNet data are
produced by choosing one image per class from the training
data, and we generate 1000 data samples randomly using the
existing method and the proposed method. As one can see,
our method shows similar or higher performances over exist-
ing data-free method, having less than 0.5% differences from
the results of ImageNet cases. As shown in VGG19, although
the existing research is hard to generalize, the method main-
tains sound performance regardless of the network structure.
3.2. Comparison of Sensitivity Metrics
To verify several metrics in weight sensitivity, we measure
the task loss by switching floating-point weights in the or-
der of layers with higher sensitivity values, where all weights
are quantized to 4-bit, and activations do not have quanti-
zation error. We denote this experiment as A32W f32,4g.
Af32,4gW32 indicates the evaluation of the metrics in acti-
vation sensitivity when switching the 4-bit quantized activa-
tion to floating-point, where weights are single-precision. Ze-
roQ [10] measures the KL divergence and [15] measures the
Euclidean distance between the original model and the quan-
tized model. ZeroQ [10] only measures the weight sensitivity.
HAWQ-V2 [16] uses the average Hessian trace and L2 norm
of quantization perturbation. We use the data generated by
the proposed method for all metrics.
Figure 3 shows the results of ResNet50 and InceptionV3(a) ResNet50 A32W f32,4g
(b) ResNet50 Af32, 4gW32
 (c) InceptionV3 A32W f32,4g
(d) InceptionV3 Af32, 4gW32
Fig. 3 . Results of sensitivity metric evaluation on ResNet50 and InceptionV3 over ImageNet dataset. A32W f32,4gis the
evaluation for weight sensitivity and A f32,4gW32 is for activation sensitivity.
Model Metric A32W f32,4gAf32,4gW32
ResNet50Proposed 1 1
L2 [15] 1.22 1.11
ZeroQ(KLD) 1.14 18.48
HAWQ-V2 1.42 2.09
InceptionV3Proposed 1 1
L2 1.04 1.50
ZeroQ(KLD) 1.69 8.06
HAWQ-V2 1.25 6.17
Table 2 . Quantitative comparison of different sensitivity met-
rics through relative area calculation of task loss graph.
over ImageNet dataset. Our sensitivity metric reliably and
quickly lowered the task loss, which means that the proposed
sensitivity metric is good at weights and activations that are
comparatively more susceptible to quantization. To express
the results of Figure 3 quantitatively, we calculate the relative
area under the curve of task loss graph considering the result
of the proposed metric as 1 and summarize it in Table 2. Our
proposed method has the smallest area in all results.
3.3. Sensitivity According to Dataset Type
To show the importance of the data in measuring the sensitiv-
ity, we evaluate the proposed sensitivity metric over the differ-
ent datasets of Section 3.1 on the 4-bit quantized network that
clipped the activation range using ImageNet dataset. We mea-
sure the task loss as in Section 3.2. The proposed dataset is the
most similar in sensitivity to the ImageNet dataset, as shown
in Table 3. Notably, the proposed data generation method pro-
vides reliable statistics similar to the original training dataset.
For InceptionV3, preprocessed ImageNet data are in the range
of [-2.03, 2.52]. However, the range of the preprocessed data
from [10] is measured as [-11.50, 11.21], while that of our
data is in [-2.35, 2.36], whose maximum value is almost simi-
lar to that of the ImageNet dataset. These similar statistics are
seen in ResNet50. The incorrect statistics of the activation
data corresponding to the original training dataset implies in-Model Dataset A32W f32,4gAf32,4gW32
ResNet50Proposed 1 1
ImageNet 0.74 1.60
Noise 1.35 6.76
ZeroQ 1.11 3.17
InceptionV3Proposed 1 1
ImageNet 1.07 2.48
Noise 1.36 3.54
ZeroQ 1.47 2.63
Table 3 . Quantitative comparison of using different datasets
in sensitivity measurement through relative area calculation
of task loss graph.
accurate sensitivity measurement [10]. Thus, our generation
method can be used for reliable sensitivity measurement.
4. CONCLUSION
In this paper, we proposed an effective data generation
method for post-training mixed-precision quantization. Our
approach is to train the random noise to generate data by using
class-relevant vectors. It is not only independent of network
structure but also provides a labeled dataset. We demonstrate
that the generated data are sufficient to ensure the quality of
post-training quantization. Furthermore, we proposed a novel
sensitivity metric, which is important to optimize bit alloca-
tion. The proposed sensitivity metric considers the effect of
quantization on other relative layers and task loss together us-
ing the gradient perturbation of the quantized neural network.
Comparisons of sensitivity metrics were made to show the
extent to which a layer with high sensitivity, measured with
sensitivity metrics of other methods, affects task loss. The
proposed sensitivity metric outperformed other metrics to
stand for the effect of quantization error. We leave optimizing
bit allocation using the proposed metric and applying to other
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