ClimbQ: Class Imbalanced Quantization Enabling Robustness on Efficient Inferences
Authors: Ting-An Chen
De-Nian Yang
Ming-syan Chen
Venue: NeurIPS 2022 Accept (2022)
Keywords: Quantization
Efficient Inference
Neural Networks

Abstract: Quantization compresses models to low bits for efficient inferences which has received increasing attentions. However, existing approaches focused on balanced datasets, while imbalanced data is pervasive in the real world. Therefore, in this study, we investigate the realistic problem, quantization on class-imbalanced data. We observe from the analytical results that quantizing imbalanced data tends to obtain a large error due to the differences between separate class distributions, which leads to a significant accuracy loss. To address this issue, we propose a novel quantization framework, Class Imbalanced Quantization (ClimbQ) that focuses on diminishing the inter-class heterogeneity for quantization error reduction. ClimbQ first scales the variance of each class distribution and then projects data through the new distributions to the same space for quantization. To guarantee the homogeneity of class variances after the ClimbQ process, we examine the quantized features and derive that the homogeneity satisfies when data size for each class is restricted (bounded). Accordingly, we design a Homogeneous Variance Loss (HomoVar Loss) which reweights the data losses of each class based on the bounded data sizes to satisfy the homogeneity of class variances. Extensive experiments on class-imbalanced and benchmark balanced datasets reveal that ClimbQ outperforms the state-of-the-art quantization techniques, especially on highly imbalanced data. 

ClimbQ: Class Imbalanced Quantization Enabling Robustness on Efficient Inferences
Ting-An Chen De-Nian Yang" Ming-Syan Chen
GGaduute Institute of Electrical Engineering, National Taiwan University, Taiwan
PInstitute of Information Science, Academia Sinica, Taiwan
$ RResearch Center for Information Technology Innovation, Academia Sinica, Taiwan tachendarbor ntu. edu. dnyangdiis sinica, edu. tw, machenEntu, edu.
Abstract
Quantization compresses models low bits for efficient inferences which has received increasing attentions. However, existing approaches focused on balanced datasets, while imbalanced data pervasive in the real world. Therefore, in this study, we investigate the realistic problem, quantization on class- daas-imbalanced data. We observe from the analytical results that quantizing imbalanced data tends obtain a large еπтоr due to the differences between separate class distributions, which leads to significant accuracy loss. To address this issue, we propose novel quantization framework, Class Imbalanced Quantization (ClimbQ) that focuses on quantization error reduction. ClimbQ
diminishing the inter-class heterogeneity first scales the variance of each class distribution and then projects data through the new distributions to the same space for quantization. of class variances after the ClimbQ process, we examine the quantized features and derive that the homogeneity satisfies when data size for each class is restricted (bounded). Accordingly, Loss) which reweights the data losses each class based on the bounded data sizes to satisfy the homogeneity of class variances. Extensive experiments class- imbalanced and benchmark balanced datasets reveal that ClimbQ outperforms the wate-of-theear quantization techniques, especially highly imbalanced data.
guarantee the homogeneity
design Homogeneous Variance Loss (Homo Var
Introduction
Convolutional neural networks (CNNs) with high computational complexity inhibits the deployment on resource- -ssuurce-limited mobile devices [1]
3 4 5]. Quantization has been studied an efficient method for reducing memory storage and accelerating inferences by compressing models low
2
10 11 [2]. The existing quantization approaches are generally developed under common assumption that the data is balanced for separate classes. However, Jlas--mbbllanee data
bits 66. 7
g 9
ubiquitous in the real world.
Despite a realistic problem, quantization on class-imbalanced data has not been studied. Therefore, in this research, we investigate the issues of quantization on Jlas-iiballanee data. The exploration results on CIFAR-IO-LT (long-tailed CIFAR-10) are depicted in Fig. T Fig. iallusstrate the data size of each class, where the largest class (indexed 0) has 50 times more data than the smallest class (indexed 9), i.e., an imbalanced dataset. We then conduct experiments quantize the class- imbalanced data by exploiting a baseline quantization approach, uniform quantization. Fig. shows that the majority classes (indexed 0 to 3) are quantized bits with only accuracy loss (compared to the 32-bit model), whereas the minority classes (indexed 8 and 9) obtain 14% accuracy drop. It indicates that the quantization result is inclined to dominated by the majority classes, which may result from the heterogeneity of class distributions as presented in Fig. Tc Fig. Tc manifests that
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Data Size
Accuracy
1k-
(5.56,77 Class Index
(4,4)
Data Size of CIFAR-IO-LT. (b) Acc. Drop under Quantization.
(c) T-SNE of Features.
Figure I: Experiment results on CIFAR- 10-LT. Fig. la presents the data size of each class in CIFAR- 10-LT flon-ttailed CIFAR-IO), 10), where the maximum class has 50x data more than the minimum class. Fig. depicts the rate of accuracy loss under quantization in comparison to the 32-bit model, where the majority classes have the least aceuracy drop, while the accuracy of minority classes decreases the most (as much as 14% accuracy drop for a 2-bit model). It indicates that the majority classes determine the quantization result due to the fact illustrated in Fig, that the majority classes not only have large amount of data but also distributed in large variations.
the majority classes always have greater data variations than the minority classes. Therefore, the quantization results are inclined to be determined by the majority classes, which may lead to large quantization error and notable accuracy loss for the minority classes (validated in Fig. [bb.
Motivated by this challenge in quantization, we propose Class Imbalanced Quantization ( (ClimbQ) scheme to reduce the differences between class distributions for a robust quantization on imbalanced data. According to the result as presented in Fig- [T that the majority classes are distributed more diversely than the minority classes, ClimbQ scales the class distributions, allowing the class variations to be closer. Afterward, we project data through the new class distributions to the same uniform space for quantization to further decrease the quantization error
To further ensure that the class distributions after quantization are homogeneous, we analyze the homogeneity of class variances based on Levene's 's hypothesis testing in the theory of statistics [3] As our contribution, we demonstrate that the data size of each class is restricted to satisfy the homogeneity of class distributions (variances) and derive the bounds of the data sizes. Accordingly. we design a Homogeneous Variance Loss (HomoVa Var Loss) to rebalance the data of each class. We place emphases on the (minority) classes with data size below the lower bound and weight down the loss of the (majority) classes with a larger data size over the upper bound to prevent the majority classes from dictating the quantization results.
Experiments demonstrate that ClimbQ+ (ClimbQ incorporated with Homo loss) outperforms the prior works on both imbalanced and balanced datasets. ClimbQ+ improves the state-of-the-ar quantization approaches by 6% for 2-bit ResNet-56, while by 12% for 2-bit MobileNet- V2 on CIFAR- FFFAR-100-LT in highly imbalanced instances.
Our contributions are summarized follows:
1. We develop novel quantization scheme, ClimbQ, that makes the first attempt to reduce the differences between class distributions for robust quantization process on imbalanced data.
2. We derive the bounds of data size for each class to guarantee the homogeneity of class variances. Accordingly, propose Homo Var Loss to reweight the loss each class in order to prevent the majority classes from dictating the quantization results.
3. Experimental results demonstrate that ClimbQ+ (ClimbQ with Homo Var loss) has superior improvements over the state-of-the-art quantization on class-imbalanced and benchmark balanced datasets, especially significant for the low-bit models on highly imbalanced data.
2 Related works
Quantization. Quantization has received increasing attentions in recent years with the develop- ment of the Internet of Things (IoT), meeting the requirements of efficient computing and storage.
We adopt the uniform quantization scheme due its efficiency when deployed hardware devices 6 5

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(a) Class Imbalanced Quantization
(c) Learning Modules
Class Distribution
x f. Mu,
Quantized CNN
Scaling & Generation
Quantization, @
Q
(b) Rebalancing for Homogeneity of Variances
FC
Q6' 1
Appropriate Class Data Size Estimation
w(n, m
Homo Var Loss, Zn
Figure 2: Overview of ClimbQ. Fig. 2a illustrates ClimbQ (CLass IMBalanced Quantization), which reduces differences between class distributions by scaling the standard deviation dı by Cg- Afterward, we project the data to the uniform space for quantization Q (detailed in Sec. 3.1). The ClimbQ algorithm is applied to each convolutional layer as shown in Fig. 2c. Furthermore, Fig. 2b, we derive the bounds n and ny that satisfy the homogeneity of class variances by examining the quantized features under ClimbQ. Accordingly, we propose Homo Var loss Lx rebalance the classes by weighting. We mainly focus on the classes with data sizes out of the bounded range (introduced in Sec. 3.2). Fig. 2c shows that the HomoVar loss Ln is computed by multiplying the weight w and data loss L for each class (illustrated as blue and green modules).
presented in
Blantization-awarr training (QAT) is branch of approaches. It mainly focuses learning clipping range [6 14 5 [3] or a data transformation [7 g III [6] for quantization with small loss of accuracy. The well-trained statistics are then stored and utilized in low-bit inferences. In contrast, Zero-shot quantization (ZSQ) has recently been proposed to incorporate the concept of knowledge distillation, transferring the information from 32-bit models to the quantized networks
17 10 18 19 20). They intend to reduce the feature differences between full-bit and low-bit models. Existing QAT ZSQ methods learned single quantization scheme for all data, implicitly assuming that the data follows identical distributions [5] 14 9 [7
TO I1 [6]. However, slass-imbalanced data realistic case which the class distributions are diverse. It visualized in Fig. Ic that the majority classes are always with greater variations and sizes, dominating the quantization results and resulting significant quantization error for the minority classes, as seen Fig. [b] Therefore, we aim to develop a robust quantization scheme on imbalanced data with reduced quantization error. In particular, this work mainly follows the QAT research instead of ZSQ which exploits additional knowledge from a floating point model and suffers from higher memory costs.
fass-imbalanced learning. Class-imbalanced problem poses great challenge in deep learning, where the minority classes with few data generally have an inferior classification performance to the Previous research has studied resampling balance the classes 21 They focus on collecting more data from the minority classes (i.e.. over-sampling) or climinating some data from the majority (i.e., under-sampling). Nevertheless, over-sampling increases memory storage and training time, whereas under-sampling raises the wver-fitting problem since models are only learned from the sampled data 28).. Another branch of class iaaassmmalanced learning approaches developed is reweighting. In contrast, they reweight the data losses of each class for balancing instead of sampling, which has no problems of memory consumption and over-fitting. Prior research suggested focusing more on the minority and assign Nevertheless, they are only concerned with class data sizes. In this research, we design reweighting loss that leverages not only the data size but also the quantized features to balance data for homogeneity of class variances with a reduced error during quantization.
majority classes (2 22
23
24
to address this issue 25
26
small losses to the majority 229 30 31
32
33
3

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3 ClimbQ
Fig. 2 introdduess ClimbQ, novel quantization scheme that considers the heterogeneity of class distributions on class-imbalanced data to reduce quantization errors. ClimbQ first scales class distributions that enables the new distributions with closer variances for quantization (depicted in Fig. 2a and detailed in Section 3.1). In Sec. 3.2, we investigate the homogeneity of class variances based quantized features. From the analytical results, satisfy the homogeneity condition (see blue modules in Fig. 2b and Fig. 2c). In Sec. 3.3, we design a HomoVar loss that reweights the data losses of each class according to the quantized data and the class data size (see green modules in Fig. 2c).
derive the bounds of data sizes that
3.1 Class imbalanced quantization (ClimbQ)
Existing works quantized data from different classes under the same process, ignoring the heteroge- neous class distributions in the real-world imbalanced data. As visualized in Fig. Ic CIFAR- LIFAR-1OLLT, the majority classes have greater variations than the minority classes, raising the concern that the quantization results incline to be bias toward the majority. As a result, the minority classes are quantized with inevitable biases that result in a significant accuracy loss (validated in Fig. Tbp. To reduce the quantization error caused by the differences between class distributions, we propose class imbalanced quantization ( (ClimbQ), as shown Fig. 2 which scales the class data obtain closer distributions (see Sec. 3.1.1) and project them to the same space for quantization (see Sec. 3.1.2).
3.1.1 Class distribution scaling and generation
According to Fig. Ic majority classes are always distributed in wide ranges, whereas minority classes are distributed in small variations. Such different class distributions cause massive quantization errors and derive a notable accuracy loss, especially for the minority classes (validated in Fig. [b). As result, we aim diminish majority class variations while enlarging minority class variations. First, we denote the network features for N data as X = {ar 22, ..., JNN and suppose that the data number for the j--th class is ni,, with k 1, 2, ..., K (in total K classes). Assume the k-th class distribution f& follows the normal distribution denoted as N(uk; og), where f4 and σr indicate the mean and standard deviation of the distribution We scale the k-th class distribution f& N N(u of o to f% N (uu, (2) where C is the scale factor defined as 2 Σ my Ak = 1,2, .... K The design reveals that the (minority) classes with less data have been scaled to larger variances. The
variances of the (majority) classes with a large amount of data are reduced. In the next subsection, we will develop an efficient quantization method based on the scaled class distributions.
3.1.2 Class distribution quantization
As illustrated in Sec. 3. 1,, data are scaled N N(pr (c) based on the class belongs to. In the following, we aim to design an efficient quantization process based on the new class distributions. Recent works adopted and efficient process, the uniform quantization, due to its simplicity and in operations on devices 6 However, since data distributions are typically
not uniformly distributed, the quantization process produces massive errors 34
35
7 5 Prior re-
search therefore proposed to learn a clipping function, quantization statistics, and other preprocessing procedures to reduce the error 7 16). Nevertheless, the proposed preprocessing procedures are complicated and necessarily require additional learning parameters. Therefore, we propose one-step transformation without additional learning parameters that allows the class data to be projected to a uniform space with minimal error under the uniform quantization scheme.
Previous research have studied that CNN weights and features follow normal distributions 5 we validate the normality the benchmark datasets in Appendix D. Therefore, in this study, we assume data follow normal distributions.
In addition,
'We adopt the distribution function of the standard normal (0, 1). instead of N (μu, (caor)²), for inference, since the class information is unknown for the testing data. Note that the main reason that utilized the standard normal that the features had been normalized through the batch normalization layer before quantization.
A uniform quantization scheme quantizes the continuous data values to the discrete values with the same intervals.

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We adopt the distribution function (namely, cumulative distribution ) as the uniform space projection function. In the following theorem according [36,, we prove that the data are uniformly distributed after the projection.
Theorem 3.1. (Proved in Appendix A.1) Let f be the continuous probability function of random variable X with its distribution function F, then F(X) = U, where U is the random variable of the standard uniform distribution.
Therefore, by Theorem 3.1
exploit the distribution function of the new class distributions f% : N (uR, (c)O)>²)) ),, vk (determined in Sec. 3.1.1), to project data Ty from the k-th class to the uniform space denoted as Dr. The projection function is then formulated as:
1
St-pa) "0E%% di, Vk- = 2,,,.... K,
(a(+))= = CLO
(1)
where Ca is the scale factor introduced and defined in Sec. 3.1.1. Eq. ① indicates that the data projection conditioned on its belonged class and the data size. After transformed to the uniform space, the data z'.- == (x(x is then processed by the following uniform quantization function:
Q(x1)) round(2 xxk)
(2)
28
where round is the rounding operation, and B represents the quantization bits.
In summary, as depicted in Fig. 2h. the data from different classes are scaled to diminish the differences between class distributions (detailed in Sec. 3.1.1) and then projected the uniform space for quantization (see Eq. ( and Eq. (2)). ClimbQ process is applied in each convolutional layer as presented as the red module in Fig. 2p, i....e. i.e. the quantized data ((2²)) output from the last
layer acts as the input zı. for the current layer. The effectiveness of the scaling on quantization error reduction demonstrated in Appendix E.
3.2 Rebalancing for homogeneity of variances
In Sec. 3.1, we proposed distribution scaling and uniform space projection to reduce the differences between class distributions for minimal quantization error. To ensure homogeneity of the class distributions after quantized through the ClimbQ process, in this subsection, we examine the variances of the quantized features from different classes based contribution is the derivation of the bounds of the data size for each class to satisfy the homogeneity design a Homo Var loss that focuses on the classes with data sizes out of the bounds (as illustrated in the green modules in Fig. 2p
Levene's 's hypothesis testing. Our main
(illustrated in the blue modules in Fig. P and Fig. 2p). Accordingly,
and Fig. P To prevent the results from being dominated by the majority classes (29 30 31 32),
intend to weight down the losses of those classes whose data size exceeds the upper bound, i.e., the majority classes, while putting more weights the losses of the minority classes with smaller data size than the lower bound.
3.2.1 Appropriate class data size estimation
For systematic analysis of the homogeneity of class distributions following the ClimbQ process (introduced Sec. 3.1), we refer to Levene's 's hypothesis testing based on the theory of statistical learning [13]. Assume there are total K classes in the dataset. Two hypotheses are included in Levene's 's testing: null hypothesis Ho of = oa = = of and alternative hypothesis Hg : of * oz; for arbitrary i # j. i, 1, 2, ..., K. The null hypothesis claims the homogeneity of variances, whereas the alternative hypothesis stands for the heterogeneity if any variance is much different from the others.
According to Levene's 's hypothesis testing, the null hypothesis is satisfied when the testing statistics W is small. The testing statistics W is formulated as the following:
(z - 22)2 (Rs 1)
SB
W
K
Zz. )²/NN k) Sw
where ZAi = |a
1 Th i= Tkil is the variation, i.c., the distance of data zns from the center of the
k-th class
5
2t.
Σ Li Zki represents the mean variation of the k-th class, and ž.. = K >K Zk-
Sx here is same the notation in Sec. 3.1 that denotes the data (features) of convolutional layers.
5

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stands for the mean variation for all data. In addition, and N individually denote the data size of the k-th class and total data size, i..., N -k=1" 71k.
The denominator of W in Eq. (3) represents the sum of the differences from the within class variations, whereas the numerator shows the total difference from the between class variations. Thus, we denote the former as Sw and the latter as SB in Eq. ( The null hypothesis is accepted when W is small, i.e., W < Fo (k- 1, NN k), where (k N' k)) is the statistics of F-distribution with degree of freedoms, k - and N k under the significance level a that manifests the statistical significance on the testing results
Derived from W < Fo (k N k). we prove in the following theorem that each class data size is restricted with a lower bound n and an upper bound n Ak = 2, .... K. It indicates that when class data sizes are not too tiny or excessively large, class variances are homogeneous (Ho is satisfied). On the other hand, when class data sizes are too small or large, significant differences appear between class variances (Ho rejected).
Theorem 3.2. (Proved in Appendix A.2) Given the definitions and the notations in Eq. (3) if W <. Fa (k 1, N k), where F& (k 1, k) is the statistics of F-distribution with degree of freedoms (K: - N k) under the significance level a, then class data size n < Tkk <
n vk = -,,2, K. The bounds n2/ n 25. E. SutCa C22/22.... E.h #u+ 42. =u))?
where C= (k- 1)- Fa (k N.N k)) Sw Σ C)tkn -2.)².
Theorem 3.2 demonstrates that if the data size of class v& is in the interval [nt. n Ho is satisfies, i.e. the class variances are homogeneous. Otherwise, Hn is rejected, which reveals that classes with out-of-range data sizes (too minor or too large) may violate the homogeneity of class variances. Therefore, in the subsequent subsection, we design a loss using the analyzed bounds to reweight the data losses of each classes to adapt the class distributions to satisfy the homogeneity.
3.2.2 Loss for homogeneity of class variances
In the preceding subsection, we derived the bounds of data sizes for each class that satisfies the homogeneity of class variances (see Eq. (3.2)). Consequently, in this subsection, we propose reweighted loss, named HomoVar loss, to reweight the classes that may exhibit significant differences from other classes. We consider three cases classes: 1) minority classes with a data size below the lower bound, 2) moderated classes with a data size within the required range, and 3) majority classes with data size exceeding the upper bound.
To avoid the majority from dominating the prediction outcomes, we primarily focus on the minority and downweight the majority (shown in Fig. [b). That is, the minority classes are penalized the greatest, followed by the middle and then the majority. In addition, we evaluate the degree of minority or majority of class, i.c. how close the class data size is to the nearest bound. Classes with minor (enormous) data size far from the derived bounds are weighted more heavily (weighted less).
According to the above insights, we design the Homo Var loss denoted as Ln in the following:
K
The
K
lin
LH
Wk
LKd
Iki),
(4)
N k=
N k=
where Lsi represents the loss the i-th data of the k-tt class, Wk 1-9 mgl
is the weight on
1-8 ""g
the data of the k-th class, 3 is a constant factor in (0, 1) that scales the losses on different classes (discussed in Appendix B). The term |ng. n is used to measure the degree of the minority and majority, where ni, namely the expected data size, is defined as the nearest bound to the data size
the actual data size ng is out of the range [nf, n. otherwise, ni 7
When the h-th class is an extraordinarily minor class, the weight wi is around 1. Moreover, wp is near to
when nę is inside the restricted range satisfying the homogeneity of class
'A larger a results in a smaller (k - 1,N - k) which is of greater evidence to support the homogeneity of class differences. In statistical hypothesis testing, the significance level alpha is typically set to 0. 0.05, or 0.001. In this paper, we adopt the significance level of 0.05. We set nf to nu instead of ny is to avoid the zero denominator in Eq.

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Table 1: Accuracy (%) on CIFAR-IO-LT under quantization. * indicates the quantization approach
fails at the imbalance ratio y,
4-hit ResNer-20
2-bit ResNet-20
4-bit MobileNet-V2 V2
2-hit MobileNet-V2
Methods
7
200
y=
50
200
7 7=
50
200
y
50
200
LLSQ 1.5
68.57
48.20
58.90
64.28
53.08
34.08
ZeroQ [1
79.37
69.40
58.70
77.75
69.25
59.10
80.05
69.85
58.93
Choi al. 17
79.69
69.74
59.42
77.80
69.36
58.46
80.13
69.98
61.04
59.25
ZAQ [00
79.81
69.71
59.98
77.72
70.03
58.79
80.15
70.82
58.49
Qimera M
79.22
69.96
61.57
53.85
39.82
34.16
9
BatchQuant I6
76,98
69.19
56.55
71.74
59.39
54.27
73.93
16.24
ClimbQ
80.72
70.90
61.31
79.25
70.33
59,44
78.91
71.77
60.98
69,94
61,74
52.47
ClimbQ+
80.58
72,28
61.69
78.88
71.73
61.06
80,24
71.82
62.10
72.51
63,41
56.89
decreases as the size T2g increases, eventually approaching - / when n& exceedingly large, i.e., a majority class. If the data size is far beyond the upper bound, we approaches zero. The results are consistent with the ideas behind the design: 1) the weights of minority classes are heavier than those of majority classes, 2) the weights increase as the class data size falls far below the lower bound, and 3) the weights reduce as the class data size exceeds the upper bound. We analyze and discuss further details in Appendix A.3.
variances. Additionally, demonstrates that
E.
4 Experiments
4.1 Experiment settings
flaas-imalaanced datasets. We evaluate the effectiveness of ClimbQ and ClimbQ+ (ClimbQ with Var loss) on the class-imbalanced datasets, Syndigit-LT (37]. CIFAR-IO-LT (38 and CIFAR- 100-LT 38). A parameter setting y determines the degree of imbalance in the datasets. 7 is the ratio between the data size of the maximum class and the size of the minimum class. For example, y 100 indicates that the data size of the largest class is 100 times larger than the minimum class. In this paper, we train at imbalance ratios of 10, 50, and 200 and validate on the balanced testing data (i.e. y = 1) to fairly evaluate the performance of each class.
Benchmark balanced dataset. Although we are primarily concerned with imbalanced data, we compare with the baseline quantization approaches on the benchmark dataset ImageNet-ILSVRC
2012 39 The imbalanced ratio is 1.77, which is close to perfectly balanced data (7 = 1).
Architectures. We use ResNets 440 and MobileNet-V2 V2 41 as the underlying architectures. ResNet is a benchmark architecture for quantization research, while MobileNet- V2 is essentially an efficient architecture with lightweight module designs that is also commonly used in image recognition.
Training. We utilize an NVIDIA Tesla V100 GPU and an NVIDIA GTX 2080Ti for implementation. ImageNet has batch size of 512, while Syndigits-LT,, CIFAR-
and CIFAR- 100-TT have 128. The maximum number of training epochs is 200. The range of the learning rate is from 0.01 to 0.1. The significance level referring to Theorem 3.2 set to 0.05. The constant factor 3 in Eq. (4) is
set to 0.999. More discussions about the setting are detailed Appendix B. Code available at ctm/ hamm//tnnancchen/CI1 ctwp//llmmmm..cmm git
4.2 Comparison results
In the following, we compare the quantization performance of ClimbQ and ClimbQ+ with QAT
g
[5
m 16 and ZSQ baselines [99 18 on the class-imblanced datasets with distinet
imbalance ratio settings
CIFAR-10-LT. The results of 4-bit and 2- quantization of ResNet-20 and MobileNet-V V2 on CIFAR-10- are presented in Table ! First, we can observe that the prediction accuracy declines with the imbalance ratio y increases. In addition, the process of low-bit quantization reduces precision.
'I this paper, model weights and features are quantized to low bits for each convolution layer. The baselines being compared have the same quantization settings.
7

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Table 2: Accuracy (%) on CIFAR-I00-LT under quantization. indicates the quantization approach
fails at the imbalance ratio y,
4-hit ResNer-56
2-bit ResNet-56
4-bit MobileNet-V2 V2
2-hit MobileNet-V2
Methods
7= 10
200
7
50
200
7
50
200
y
50
200
LLSQ 1.5
34.46
26.23
11.60
ZeroQ [8
5.74
4.92
1.38
5.47
5.27
1.27
53.79
27.29
20.91
25.91
15.89
13.56
Choi al. 17
6.25
4.09
5.03
1.18
1.00
53.84
36.38
28.30
26.51
15.27
12.38
ZAQ 10
18.81
+
52.86
35.70
27.15
Qimera M
48.64
34.44
8.60
2.98
BatchQuant I6
41.50
28.91
22.91
37.24
27.03
22.17
41.50
23.77
21.48
35,80
23.79
10.69
ClimbQ
50.04
34.30
25. 64
47.15
33.85
23.91
53.74
31.47
22.58
41.50
22.02
18.53
ClimbQ+ 52.13
35.10
27.46
49.12
34.79
27.39
55.62
36.89
28.47
43.96
33,18
25.45
In particular, most of the compared quantization approaches have significant accuracy loss for low bit models in highly imbalanced cases, since they quantize on imbalanced data (with heterogeneous class distributions) using the same quantization functions or distillation method, which results in a huge quantization error and significant loss of accuracy. contrast, our Climbb and ClimbQ+ quantization schemes are proposed to diminish the heterogeneity in different classes (introduced in Sec. 3). Table [ demonstrates that our proposed methodologies can effectively reduce quantization error and accuracy loss, especially on the lightweight architecture design, MobileNet-V2. ClimbQ increases the accuracy of baselines under all studied conditions. In addition, ClimbQ yields a performance increase of up to 10% for 2-bit MobileNet-V V2 with 7 10 and can successfully implemented on the highly imbalanced cases = 50 and 200. ClimbQ+ can further improve the performance of ClimbQ by as much as 4.4% in absolute accuracy. The exceptional performance is result of attempts to reduce the heterogeneity of class variances using quantized features and class data sizes (see the evaluation of homogeneity and the proposed Homo loss in Sec. 3.2).
CIFAR- RNFAR-100-I4 10---....T We also study the efficacy of ClimbQ and ClimbQ+ on CIFAR- 100--- which contains 10 times many classes as CIFAR-10-LL) Due to the increasing difficulty of the recognition task, we employ a deeper ResNet-56 model instead ResNet-20 for quantization. Table 2 reveals that the accuracy of the compared approaches degrades significantly under highly imbalanced circumstances. ClimbQ+ enhances the performance of the BatchQuant by 5% to 12% accuracy on 2- ResNet-56 and increases 8% to 15% for 2-bit MobileNet- V2. validating the effectiveness of the ClimbQ process (scaling the class variances as introduced in Sec. 3.1) and the Homo Var loss (rebalancing the data size of each class shown in Sec. 3.2) on quantization error reduction.
Syndigits-LT and ImageNet. We further evaluate and discuss the performances of ClimbQ and ClimbQ+ on the imbalanced dataset Syndigits-LT and balanced benchmark ImageNet in Appendix C. ClimbQ+ on Syndigits- in the highly imbalanced case, y = 200, achieves remarkable 80.55 accuracy for 2-bit ResNet-20, i.e., 22% accuracy improvement over the best baseline result evaluated (only 58.75% accuracy). Moreover, ClimbQ+ on 2-bit MobileNet-V outperforms the state-of-the-an BatchQuant with 26% increase in accuracy. In addition to the imbalanced datasets, we compare our quantization methods against those previously proposed on the balanced benchmark ImageNet. ClimbQ+ outperforms the previous studies by approximately 3% in terms of accuracy. More details are discussed in Appendix C.
Ablation study on rebalancing strategies
In Sec. 4, demonstrated the superior performance of ClimbQ (introduced in Sec. 3.1) and ClimbQ+ (described in Sec. 3.2) over the baseline quantization approaches, which is particularly with significant improvement for low-bit and lightweight models in extremely imbalanced scenarios. In this section, we compare ClimbQ+ to other effective rebalancing methods") Table demonsttrtss that ClimbQ+ has outstanding performances compared the previous research on rebalancing data sizes. ClimbQ+ improves absolute accuracy by 4.4% to 22.4% for the 2-bit model on CIFAR-10-1 -O-LT with 7 50 and by 2.6% to 25.3% accuracy with 7 200. The extraordinary performances of ClimbQ+ verifies that the Homo Var loss introduced in Sec. 3.2 can effectively reduce more quantization errors
'The rebalancing methodologies are implemented under proposed ClimbQ quantization process on the imbalanced data to compare with reweighted loss, the Var loss (designed in Sec. 3.2).
8

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Table Accuracy (%) of rebalancing approaches on ClimbQ based low-bit ResNet-20. indicates
the quantization approach fails at the imbalance ratio 7,
Syndigits- LT (4-bit)
Syndigits-LT (2-bit)
CIFAR- I0-LT (4-bit)
CIFAR-IO (2-bit)
Methods
y=
10
50
200
10
50
200
7 ==
50
7
Focal 29
96.05
88.20
76.35
95.20
87.15
76.05
80.52
66.69
55.80
78.27
55.96
37.69
CB 30
96.10
88.45
76.45
95.50
87.60
74.80
76.31
69.70
60.46
65.62
59.94
51.69
LDAM [2
95.25
86.85
78.15
94.80
86.35
76.90
80.48
70.68
56.01
78.65
49.38
37.21
Causal
96.20
88.65
79.70
95.40
87,90
78.50
76.60
69.81
60.05
73.44
67.33
58.44
LADE 31
95.45
85.65
72.10
95.15
85.45
71.80
80.69
64.99
49.56
78.88
60.24
47.45
ClimbQ+
96.45
89.55
80.70
95.75
88,90
80.55
80.58
72,28
61.69
78.88
71.73
61.06
and accuracy loss owing the rebalancing strategy that takes the homogeneity of the class variances (variances quantized features) into a consideration, rather than only leveraging the class data sizes.
6 Discussion with potential impacts and limitations
Potential impacts. ClimbQ with the HomoVVr loss aims to rebalance the classes for homogeneity of class variances. To prevent the majority classes from dominating the quantization results, the rebalancing strategy focuses primarily on the minority classes while scaling down the weights on the majority classes for the homogeneity of class variances, which implies that the prediction results of the minority classes are more dependent on the small amount of data. Accordingly, they may be much easier to be attacked with notable performance degradation. However, recent research has developed the defense strategy against the attacks on imbalanced data 42 which can also be applied to ClimbQ.
Limitations. ClimbQ mainly focuses on the dlas-imbalanced data with different class data sizes and distributions. However, we believe that there are other scenarios of heterogeneous data the real world that have not been fully studied. As a result, we will explore and study more research issues of quantization on heterogeneous data in our future works.
7 Conclusion
In this paper, we study a realistic but not fully studied research problem of quantization on class- imbalanced data, where the differences in class distributions lead to huge quantization error in the minority classes. To address this issue, we propose ClimbQ to scale the class distributions to diminish the differences between classes for quantization error reduction. Morcover, analyze that the homogeneity of class variances under ClimbQ process subjected to restricted class data sizes. Therefore, further propose HomoVar loss rebalance the classes based the derived bounds of data sizes to satisfy the homogeneity. Experiments reveal that ClimbQ/Climb@+ outperforms the baseline quantization and rebalance techniques, validating the effectiveness quantization error reduction by diminishing the differences in class distributions.
Acknowledgment
This work was supported in part by the Ministry of Science and Technology, Taiwan, under grant MOST 111-2223-E-002 -006 and MOST 111-2221-E-002 1135 -MY3.
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Checklist
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Did you include the license to the code and datasets? [Yes] See Section ??.
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1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Section 6.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section and Appendix D.
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes] See Section
4.1.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section
(c) Did you report error bars (e.g., with respect the random seed after running experi-
ments multiple times)? [No] We report the best result within 200 training epochs.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4.
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12


Reviews:
[{"review_id": "j-PDzOjdQZ2", "reviewer": "Reviewer_Xgjj", "summary": "The authors investigate quantization of (deep neural network) model parameters in class-imbalance settings. This is achieved by scaling the variance of of each class separately, which is then mapped to a common distribution for quantization. Further, a loss function with class reweighing is used to satisfy the homogeneity of class variances. Experiments on four datasets (Synddigit-LT, CIFAR-10-LT, CIFAR-100-LT, ImageNet-ILSVRC) demonstrate state-of-the-art results for quantized models, specially, on highly imbalanced data.", "questions": "Though the assumption of variance differences between classes of different sizes is reasonable and can be demonstrated in practice in multiple ways, using t-SNE is perhaps not the best way considering that due to the minimum distance specification in t-SNE, a larger dataset will occupy a larger space regardless of variance. Moreover, being a local linear embedding technique, distributional variance are not necessarily well represented. Can the authors justify or use a different visualization technique?\n\nIn Section 3.1.1 the authors claim that differences in class distributions cause massive quantization error, however, it is not justified or demonstrated.\n\nThere seems to be a problem in Figure 2, by which N of N(\\mu_k,sigma_k) is not shown.\n\nThe authors argue that data follows a normal distribution, which is demonstrably not true in general practical settings (without data transformation). Though it is possible that the authors are referring to model parameters and features, the beginning of Section 3.1.2 clearly states that data x is assumed to be Gaussian, though later it is stated that x are features. Please clarify by stating the notation earlier.\n\nWhat happens in practice when beta -> 1 (0.999 < beta < 1)?\n\nDid the authors consider ClimbQ with a simple inverse probability weighting (1/p_k, where p_k is the proportion of samples in class k)?", "limitations": "The authors address the impact and limitations of the proposed approach in Section 6.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\nThough the variance scaling and quantization are relatively straightforward, the weighting approach based on hypothesis testing offers a different and interesting angle to class weighting schemes. The experiments are extensive, consider multiple datasets, imbalance ratios and importantly, also consider existing class weighting approaches, which not originally intended for quantization, can be used within the ClimbQ framework.\n\nWeaknesses:\nThough the paper is relatively well organized, it is difficult to follow, for instance, N(.,.) is used before being defined, x is introduced as data to then being extended to also the features from a deep learning model, there is abuse of notation (e.g., F(X) a distribution is equated to a random variable U), x' is not formally defined, etc.\n\nSomething that is never explained or discussed is why if the data (including features) is being scaled during training, how is the scaling being applied during testing and if not necessary, why is it the case?\n\nPost-rebuttal: increased rating after detailed clarification from the authors.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "W7jtW_X8zpp", "reviewer": "Reviewer_Q3g7", "summary": "The authors propose ClimbQ quantitzation for efficient inference in\nthe context of class-imbalanced problems. The majority (minority)\nclasses have larger (smaller) variance and they reduces (increase) the\nvariance.  To project onto a uniform distribution, they use a\ncumulative distribution function, which is then quantized.  Based on\nLevine's hypothesis testing on homogeneity in variances, they show a\nlower and upper bound of class data size to reject/accept the null\nhypothesis (homogeneous variances).  Using the bounds, they divide the\nclasses into: minority, moderated, and majority classes.  They then\ndesign the HomoVar loss, which weight minority classes more and\nmajority classes less.\n\nEmpirical evaluation on 3 imbalanced datasets indicates the proposed\nmethod generally outperform 6 existing methods.\n", "questions": "lines 186-188 vs lines 194-195: the null hypothesis is rejected when\nthe class size is NOT too small/large (lines 186-188), but it is\nrejected when the class size is too small/large (lines 194-195).\n\nEq 4: more motivation on how w_k is designed would be beneficial,\nparticularly the intent for the numerator and denominator.\n\nline 170:  nominator -> numerator ?\n", "limitations": "Limitations and negative societal impact were discussed.\n", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The problem of quantization for class-imbalanced problem is\ninteresting.\n\nThe proposed projection onto an uniform space and HomoVar loss are\ninteresting. They derived the upper and lower bounds of class size\nwith respect to Levine's hypothesis testing on homogeneity in\nvariances.\n\nEmpirical evaluation indicates the proposed approach generally\noutperform existing methods.\n\nThe paper is generally well written.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "PP3ESY3FlcW", "reviewer": "Reviewer_h4jR", "summary": "This paper investigates the issue of quantization on class-imbalanced data. The key observation in this work is that quantizing imbalanced data inclines to obtain a large error due to the differences between separate class distributions, which leads to a significant accuracy loss. In order to tackle this issue, this work proposes ClimbQ, a new framework that focuses on diminishing the inter-class heterogeneity for quantization error reduction. ClimbQ first scales the variance of each class distribution and then projects data through the new distributions to the same space for quantization. To guarantee the homogeneity of class variances after the ClimbQ process, we examine the quantized features and derive that the homogeneity satisfies when data size for each class is restricted (bounded). Accordingly, ClimbQ embeds a new Homogeneous Variance Loss (HomoVar Loss), which reweights the data losses of each class based on the bounded data sizes to satisfy the homogeneity of class variances. Extensive experiments on class-imbalanced and benchmark balanced datasets reveal that ClimbQ outperforms the state-of-the-art quantization techniques, especially on highly imbalanced data", "questions": "- What are the benefits (and/or downsides) of ClimbQ over other mixed-quantization schemes?\n- Can you comment on potential applicability of your approach on post-training quantization?\n- Can you comment on potential impacts of your approach on the inference performance?", "limitations": "- I assume the work requires the knowledge of the distribution on the full dataset (which may limit the practicality of this approach since it may require multi-party sharing).\n- I am concerned about the potential implications in terms of the inference performance.\n- I would suggest to make the scope of your approach clear upfront.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Important problems and the insights are valuable\n- An end-to-end solution to address the class-imbalanced issues within quantization\n- Evaluation results are sufficient to justify the effectiveness of the proposed approach\n- Source codes available\n\nWeaknesses:\n- Limited novelty (the solution donot add new insights on class-imbalanced quantization)\n- The proposed solution seems restricted within quantization-aware training", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "18oY5L2-8gI", "reviewer": "Reviewer_bnA4", "summary": "The authors propose better quanitization techniques for class-imbalanced datasets. \n\nSpecifically, propose \n1. ClimbQ - scale the class distributions such that rare classes have higher variance, followed by bucketization of CDF\n2. ClimbQ + homovar - a weighted loss function inspired from Levene’s hypothesis testing.\n\nThe authors compare the proposal against related techniques on 3 datasets, mostly by simulating different levels of imbalance.\n", "questions": "- Have the authors considered plugging in eq(4) to other competing approaches?", "limitations": "N/A", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths.\n---------\n\n- The proposal is well-motivated, simple and shows strong experimental results.\n- The paper is also well-written.\n\nWeakness\n--------\n- The generation of simulated data through γ is a bit unconvincing since it was not clear how exactly was the imbalance simulated.\n  Having said that, I do think results in Appendix#3 sound convincing.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept