Understanding the Unfairness in Network Quantization
Authors: Bing Liu
Wenjun Miao
Boyu Zhang
Qiankun Zhang
Bin Yuan
Jing Wang
Shenghao Liu
Xianjun Deng
Venue: ICML 2025 poster (2025)
Keywords: model quantization
supervised learning
fairness
Primary area: general_machine_learning->supervised_learning

Abstract: Network quantization, one of the most widely studied model compression methods, effectively quantizes a floating-point model to obtain a fixed-point one with negligible accuracy loss. 
Although great success was achieved in reducing the model size, it may exacerbate the unfairness in model accuracy across different groups of datasets.
This paper considers two widely used algorithms: Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT), with an attempt to understand how they cause this critical issue.
Theoretical analysis with empirical verifications reveals two responsible factors, as well as how they influence a metric of fairness in depth.
A comparison between PTQ and QAT is then made, explaining an observation that QAT behaves even worse than PTQ in fairness, although it often preserves a higher accuracy at lower bit-widths in quantization.
Finally, the paper finds out that several simple data augmentation methods can be adopted to alleviate the disparate impacts of quantization, based on a further observation that class imbalance produces distinct values of the aforementioned factors among different attribute classes. 
We experiment on either imbalanced (UTK-Face and FER2013) or balanced (CIFAR-10 and MNIST) datasets using ResNet and VGG models for empirical evaluation.

Understanding the Unfairness in Network Quantization
Bing Liu 123
Wenjun Miao|
Boyu Zhang'
Qiankun Zhang Xianjun Deng'
512 Bin Yuan
1345
Jing Wang Shenghao Liu'
Abstract
1. Introduction
In recent years, with the advancements in computer perfor- mance and the maturation of data processing technologies, deep neural networks have made significant strides in fields such as computer vision and natural language processing. achieving impressive results. However, high computational time consumption and large memory overheads pose sig- nificant challenges to the efficient implementation of deep neural networks on resource- limited devices. To address these challenges, neural network quantization (Huang et al., 2024; Jha et al., 2024; Gholami et al., 2022) is one of the highly effective methods to reduce the power and latency of neural network inference.
Network quantization, of the most widely studied model compression methods, effectively quantizes a Noating-poin model to obtain a fixed- point one with negligible accuracy loss. Although great success was achieved in reducing the model size, it may exacerbate the unfairness in model accuracy across different groups of datasets. This paper considers two widely used algorithms: Post- Training Quantization (PTQ) and Quantization- Aware Training (QAT), with an attempt to under- stand how they cause this critical issue. Theoret- ical analysis with empirical verifications reveals two responsible factors, as well as how they in- fluence a metric of fairness in depth. A compar- ison between PTQ and QAT is then made, ex- plaining an observation that QAT behaves even worse than PTQ in fairess,, although it often pre- serves a higher accuracy at lower bit-widths in quantization. Finally. the paper finds out that several simple data augmentation methods can be adopted alleviate the disparate impacts of quantization, based further observation that class imbalance produces distinct values of the aforementioned factors among different attribute classes We experiment on either imbalanced (UTK-Face and FER2013) or balanced (CIFAR- and MNIST) datasets using ResNet and VGG models for empirical evaluation.
To achieve these savings, quantization stores weights and activation tensors as low-bit fixed-point numbers (e.g- 4 or 8-bit) instead of original 32-bit floating- ppintt representation. This greatly reduces data storage requirements, as well as the size and energy consumption of MAC operations, speed- ing up network execution. There are generally two main classes of algorithms: Post-Training Quantization (PTQ) (He et al., 2024; Yao et al., 2022) and Quantization-Aware Training (QAT) (Xie et al., 2024; Nagel et al., 2022). While PTQ quantizes the model after training and requires no re- training, QAT requires fine-tuning and access to training data. Notably, Nagel et al. (2021) show that both methods do not suffer significantly in terms of model accuracy when compared to their original Roating-poin counterparts.
Although quantization causes little degradation to overall accuracy on a test set, previous studies (Hooker et al., 2019; 2020) observe that disproportionately high errors may ap- pear among different groups of datasets. As first glimpse, experiment facial recognition task with UTK-Face dataset (Zhang et al., 2017) and VGG19 model (Simonyan & Zisserman, 2014), focusing on the accuracy of quantized models by PTQ and QAT from 32-bit to 4-bit.
'School of Cyber Science and Engineering, Huazhong Univer- sity of Science and Technology, Wuhan, China Keyy Laboratory of Cyberspace Security, Ministry of Education, Zhengzhou, China 3 HHubei Key Laboratory Distributed System Security, Wuhan, China "Songshan Laboratory. Zhengzhou, China "Visiting re- searcher with the Lion Rock Labs of Cyberspace Security, CTIHE, Hong Kong, China "School of Software Engineering, Huazhong University Science and Technology, Wuhan, China. Correspon- dence to: Qiankun Zhang <qiankun @ huut..dd..en>.
We summarize our key observations in Figure as follows:
While models are unfair in accuracy before quantiza- tion among different groups of individuals, quantiza- tion exacerbates such unfairness as bit-widths get lower. For example, 4-bit QAT barely decreases accuracy in group White, but group Others suffers a dramatic drop.
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Understanding the Unfairness in Network Quantization
0.9
0.7
0.7
Others 7.0
15.0%
Accuracy
Accuracy
0.5
0.5
42.0%
Indian 17.0%
White
Black
Black
0.2
Indian
0.2
Indian
19.0%
Asian
Asian
Others
Others
0.0
0.0
foot 32
Int 16
int
int
flont 32
int 16
int
int 4
(a) PTQ
(b) QAT
(c) Percentages of five samples groups
Figure 1. Experiments on UTK-Face dataset on the accuracy of cach subgroup of individuals using ResNet50 for an ethnicity classification task. Both PTQ and QAT are evaluated as bit- get lower.
Though QAT has a better guarantee of accuracy than PTQ. PTQ behaves better in fairness preservation. For example, from 32-bit to 4-bit, accuracy in group Others decreases with a smaller percentage PTQ than QAT.
the gradient norm of the group loss function and (2) the trace of the group loss function's Hessian matrix, regardless whether PTQ or QAT is used,
class imbalance induces distinct values of both factors among classes. Both the gradient norm and trace of the Hessian matrix increase as the size of subgroup decreases, indicating a positive correlation between the two factors,
These observations should receive great attention, because such facial recognition tasks have been deployed in some lisuuree-lmiitd devices, e.g., mobile phones or access con- trol systems, and they could potentially harm fairness and privacy. This constitutes our biggest motivation in this paper for better understanding and elimination of such unfairness caused by quantization for equality protection and privacy preservation. An intuitive understanding seems not difficult: quantization involves loss of parameter information by rep- resenting weights and activation tensors as -wwpreciiion fixed-point numbers, and thus for underrepresented groups, whose available feature information is already relatively lim- ited, the loss of parameter information can further prevent accurate learning of their features. For an in in-depth study, our paper makes a step toward answering the following questions:
the unfairness caused by QAT is more severe than PTQ. because besides being influenced by the above two factors separately, QTA is additionally influenced by their interactions.
Based on the theoretical analysis, we empirically validate our findings on different datasets (UTK-Face, FER2013, CIFAR- 10 and MNIST) and models (ResNet and VGG). Fi- nally, to mitigate the unfairness caused by quantization and validate our main findings that balanced dataset is crucial for fairness preservation, a natural idea to utilize data augmentation techniques for training data. Two simple but effective augmentation methods are empirically evaluated for both PTQ and QAT.
Our Research Questions
What are the factors that exacerbate the unfairness in model quantization?
2. Related Work
How do these factors influence the disparate pro- portions of accuracy degradation among different groups of data?
As two main keywords of this paper, both quantization (Jha et al., 2024; Yang et al., 2024b; Gholami et al., 2022; Nagel et al., 2021; Jacob et al., 2018) and faimness (Caton Haas, 2024: Lalor al., 2024; Zhang et al., 2024a; Mchrabi et al., 2021) are well studied separately. For quantization, exten- sive efforts have been devoted to improving quantization performance using either QAT (Liu et al., 2023; Nagel et al., 2022; Esser et al., 2020; Choi et al., 2018) or PTQ (Xiao et al., 2023; Frantar & Alistarh, 2022; Wei et al., 2022; Li et al., 2021). For fairness, studies vary in research fields, for example, federated learning (Badar et al., 2024; Li et al.,
How can we mitigate such unfairness to address the effect these factors?
We present a theoretical analysis in disparate impacts of both PTQ and QAT on models' fairness separately. Our main findings are:
Our contributions.
accuracy disparities come from two main factors: (1)
2

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Understanding the Unfairness in Network Quantization
2019), graph neural networks (Yang et al., 2024a; Dong al., 2022), generative adversarial networks (Zhang al., 2024b; Xu et al., 2018), and etc.
in the realms of QAT and PTQ.
Building upon the experimental observations mentioned above, this article delves deeper into the specific factors behind the degradation of model faimess caused by neural network quantization and provides an effective mitigation strategy to alleviate this unfairness.
Our work falls under the broad umbrella of studying the so- cial impacts of ML techniques. Much more related to ours, previous studies have observed through empirical experi- ments that various neural network compression techniques can cause unfairness in different learning tasks, where the classification accuracy of certain categories may be more affected than others (Jha et al., 2019; Joseph et al., 2020). Most of these observations lie in CV tasks. Hooker et al. (2020) use visualization methods to study the disparate bi- ases introduced by quantization and pruning on different datasets; Tran et al. (2022) elucidate the theoretical factors that exacerbate model unfairness due to pruning in face recognition classification tasks and proposes a mitigation strategy; Hooker et al. (2019) compare the different fairness impacts of pruning and quantization and find that high lev- els of pruning incur a far higher disparate impact than is observed for the quantization techniques; and Blakeney et al. (2021) propose two simple yet effective metrics, Combined Error Variance (CEV) and Symmetric Distance Error (SDE), quantitatively evaluate the induced bias prevention quality of pruned models and demonstrate that knowledge distilla- tion can mitigate induced bias in pruned neural networks, even with imbalanced datasets. The faimnsss impact of com- pression has also been evaluated in NLP tasks. For example, Du et al. (2021) and Xu et al. (2021) measure the robustness of compressed large language models based on experience, while Ahia et al. (2021) study how compression schemes affect data restriction mechanisms. Xu & Hu (2022) in- vestigate method to improve fairness by compressing the generated language model.
3. Preliminaries
This section presents necessary background on model quan- tization, a formal definition of our research problem and the metric we use for fairness measure.
3.1. Quantization and Error Bound
Signed symmetric uniform quantization. In this pa- per, we focus on quantizations only on weights w E [min Wmaz] rather than activations. Since the distribu- tion of the neural network parameters is usually symmetric about O (Glorot & Bengio, 2010), we assume Wmin widely-used signed symmet-
W maz. Thus, we base ric uniform quantizer, which is parameterized by a scale @ (0 < 8 < 1) which specifies the step size of the 2--1-1 quantizer, and the bit-width b. It maps real-valued vector w to an integer-valued vector wine by '
W (1 Wint clamp(1
1--2--11-- 1-2-11200-111 -11-11).
a w w<a;
(I)
where clamp w; a, c)
w a a w Sc;
w>c
Note that Eqn. (I) maps floating-poin to integer 0. Floating-point weight vector w stored as integral Wint- An upproximation of w, denoted as w. can be de-quantized from Wint by:
Mitigating unfairness through data augmentation is a widely used strategy, particularly effective in addressing accuracy disparities arising from data imbalances. For datasets en- compassing protected attribute, Sharma et al. (2020) de- fine an "ideal world dataset" as data where different groups within the protected attribute attain the same label, irrespec- tive of other feature values. By implementing this data aug- mentation technique, it effectively reduces bias in line with two key fairness metrics: statistical parity difference and average odds difference. Furthermore, to improve the gener- alizability of fair classifiers, Mroueh et al. (2021) propose "Fair Mixup", data augmentation strategy for imposing the fairness constraint. In particular, they show that fairness can be achieved by regularizing the models on paths of interpo- lated samples between the groups. However, distinct from the aforementioned methods, based on our theoretical and empirical findings, we extend data augmentation techniques geometric transformation and random erasing (Zhong al., 2020) by empirically demonstrating the utility of data aug- mentation in mitigating bias in quantized models, especially
W S Wint RJ w
(2)
Combining Eqn. (1) and Equ. (2) gives a quantization func- tion q from to w:
e g(w;s, b) = s- clamp _ 20 2--1--111 1). (3)
Quantization error. Eqn. (3) indicates certain error, de- fined as Au - www between w and w after quantization. The error comes from two parts. One comes from the 1-1 operator, lying within range of 81, 1 1 (Nagel et al., 2021) for each component i of w. The other comes from the clipping error, but our quantization scheme does not introduce this error, as the quantization factor s = 2 7
LL1 is the cound-to-nearest operator.
3

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Understanding the Unfairness in Network Quantization
ensures that the quantized integers lie between - and 2b-1 -1. Thus the &2 norm 2 of Aw be upper bounded
vary among all protected attributes a E A? (3) Do QAT
and PTQ behave differently with respect p 2 (4) Any mitigation strategies can be proposed to minimize 4 DD)
by:
1 Awwl < 2V V
The following sections address all these issues.
(4)
where n is the number of dimensions of w and 8moz is the largest scale used in the quantization among all parameters. See Appendix A. for detailed proof of Eqn. We point out that readers may easily verify that s is inversely pro- portional to b, meaning that quantization to lower bits will induce larger errors.
4. Fairness in Post-Training Quantization
In this section, we discuss the degradation of fairness caused by PTQ, which takes a pre-trained 32-bit foating-poim network and converts it directly into fixed-poine network without fine-tuning. Recall that in Eqn. (1) the weight vector w is first quantized into Wint, and then calculations are based on the approximation w defined in Eqn. (2), inducing an error bound of 2 3Vnsmaa the loss function é is twice differentiable, e.g., MSE loss, To see what factors influence the excessive loss G(a) of specific group E A, the following upper bound" 3 is useful:
3.2. Problem Definition and Fairness Metric
Eqn. (4). We assume that
Empirical risk minimization (ERM). We consider a classification task over a dataset D that learns a classifier fw X y' parameterized by w using ERM. D consists of m individual data points (æ;, di 3/4 for each i E [m]] drawn i..dd from an unknown distribution. I; E X' repre- sents an input feature vector, 04 E A represents a (private or protected) attribute of subgroups, and y/ E y represents the label. We take the facial recognition task on UTK-Face in Figure as an example for explanations. 2; a face photo of an individual, a; is the ethnicity (White, Black, Indian, Asian, and Others) of each individual, and y: is also the label of ethnicity attributes. (In this task, we let A = y, but they are not necessarily identical.) w is an n-dimensional vellev vector, and is trained by:
Theorem 4.1. Let be twice differentiable loss function and consider w" is quantized to low bits using PTQ. The excessive loss for group a E AA is upper bounded by:
g(a) - v /nsmaz -19
+
ns max
-T(HH
2 + +0 (111011)))
8
(8)
where gw- = Vuw(ww: Da) is the vector of gradient associated with the ERM function C evaluated at w and computed using group data Da, Tr Tr(HD== ) is the trace of the Hessian matrix HD [uw(""; Da) of the ERM function L, at the optimal parameter vector w", computed using the group data Da. and Aw" is the quantized error of the optimal parameter vector w..
w- = arg min L(w; D) arg min w
Σ (fu((z)) U.) m
1
m
(5)
where y + R4 is a non-negative loss function.
The main ingredient of our proof for Theorem 4.1 is a second-order Taylor expansion of the objective function L at w" with the assistance of several inequalities and the consistency of lg norms between matrices and vectors.
Metric of fairness. To measure the fairness impacts caused by quantization, first recall that w" can only be approximated as in Eqn. (3), the error leads to difference between risk functions, which is called an excessive loss. That is, for each group E A, we define:
Relationship to (DD). In Theorem 4. 1, except for a neg- ligible term, G(a) is related to the sum of: (1) the product of error bounds determined by Smaz and the gradient norm Ig@:ll || for group Da: (2) the product of a term positively correlated to error bounds and the trace of the Hessian ma- trix Tr ::) for group Da- Let's consider a data group Da with protected attribute a. If the corresponding gra- dient norm 11 and trace of Hessian matrix Tr Tr(H.) are larger than other attributes, G(a) grows faster as the bit-width b gets lower (because 8maz X tt). Conversely, if Di I| and P: are small, (a) is not that sensitive to Smaz (or b). As a consequence, as the quantization bit- width decreases, the gap in G(a) values between different groups will further widen, leading to increase in yo( (D)
G(a) L(i;; Da) L(w"; Da),
(6)
where D. denotes the subset of containing exclusively samples whose group attribute a; a. w denotes the quan- tized model parameters while w" = [(wm D). Further, fairness can be measured by the largest gap of ex- cessive losses among all pairs of protected attributes in A:
xp D) = max EA
(G(a) G(a').
(7)
Eqn. (7) gives our metric of fairness. Our main goal in
this paper is to study: (1) What factors are responsible for quantized models with (DD 0? (2) Why does G(a)
Unless stated otherwise, ||- means &2 norm in this paper.
'All proofs this paper are presented in Appendix A.

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Understanding the Unfairness in Network Quantization
Table Fairness metric yp across various quantization methods and bit-widths different models and datasets. higher yp indicates more severe unfairness. Note: ()D) are presented percentages due to their originally small magnitudes for improved clarity.
(D (%)
Quantization Method
Bit-width
ResNet50
VGG19
ResNet18
UTK-Face
FER2013
UTK-Face
FER2013
CIFAR-10
MNIST
int 16
2.4
2.9
11.2
0.6
0.9
0.8
PTQ
int 8
3.6
6.9
12.4
1.5
1.0
int 4
18.8
13.4
54.6
50.7
1.3
1.2
int 16
3.3
3.6
11.7
1.4
1.1
1.0
QAT
int 8
6.0
9.9
18.9
2.9
1.3
1.1
int 4
25.4
44.4
84.1
65.7
1.5
1.4
and consequently exacerbating unfairness. The results are consistent with those shown in Table 1.
of subgroups, i.e., Dal- Details will be discussed in the rest of this section.
Our experimental results echo the arguments above. 4 We set up experiments using ResNet50 on the UTK--aaee dataset for an ethnicity classification task. Experiments in Fig- ure I(a) and Figure 2 are towards the relationship between the gradient norm, the trace the Hessian matrix, and the model accuracy, for five demographic attributes and their corresponding subgroups, at three different bit-widths in PTQ. Figure I(a) and Figure 2 illustrate Theorem 4.1 in the following way 5: (1) Consider specific bit-width, for example, int 8. While the gradient norms and the traces of the Hessian matrices are larger over different groups, e.g., comparing group White and Indian, there an opposite numerical relationship accuracy; (2) Consider a specific demographic group, for example, group White. While its gradient norm accounts for the smallest percentage, its accu- racy decreases the most slowly. Similar phenomena happen in traces of the Hessian matrices. Based on Theorem 4.1 and the observations from Figure ((a) and Figure 2, we conclude our main findings as follows:
The effeet of gradient norms. The next lemma provides an upper bound for l revealing the factor contribut- ing to the disparity in gradient norms among different groups.
Lemma 4.2. For given group Da, with protected at- tribute E AA {ai, a2,, aua), its gradient norm sociated with the ERM function L evaluated at w can be upper bounded by:
k
11g 1|
1
Σ
Day1
Day
(9)
2)D
j=1
where i [k] and = |1| 2.
Corollary 4.3. Consider groups and in A. If Dal Do). then ll @0
Lemma 4.2 associates the gradient norm of a specific group with its size, followed by direct corollary when A only con- tains two subgroups. Figure illustrates Corollary 4.3 by re- D., vealing the relationship between gradient norms ||g and sizes of groups Da., on the UTK--Face dataset. There exists strong trend between decreasing group sizes and increas- ing gradient norms for such groups, i.c.. ll x [D..TT This demonstrates that groups with smaller sizes of data points tend to have larger gradient norms than larger groups and vice-versa, leading to severer unfairness for underrep- resented groups. The observation is actually not surprising because of an intuitive understanding: when model con- verges at local optimal, it learns little features from disadvantaged group, inducing a low accuracy on this group. Now if we fine-tune the model individually on this disadvan- taged group, the loss large at the beginning and declines sharply, indicating a large norm the corresponding gradi- ent in this group
Takeaway 1:
In post-training quantization, gradient norms and traces of the Hessian matrices should pay for the exacerbation of unfairness. As bit-widths get lower, the accuracy after quantization on a demographic group with larger values of the two factors drops more dramatically.
Taking a further step, we introduce that both | and p:) are negatively correlated to the size of the dataset
"We only present the accuracy results for training on UTK- Face dataset using ResNet50 in the main body of the paper for illustration, and leave similar experiments other datasets and models Appendix C.3.
For a clear presentation, both gradient norms and traces of the Hessian matrices scaled proportion of the sum among all demographic groups in this paper.
5

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Understanding the Unfairness in Network Quantization
50.9
with 10 groups and 10% of the total for each group. Fig- ure 4(a) in Appendix C.1 reports results and shows that as the bit-width gets lower in PTQ, accuracy gaps between 10 groups on the CIFAR-I1 10 are relatively stable, while they grow larger on UTTK-Face in Figure (a). As summarized in Table I, the deterioration of unfairness on the two datasets is as follows: on CIFAR-10, R-00, the increase in up( () minimal, rising from 0.9% to 1.3%, whereas on the UTK-Face dataset, the increase in yp D) is much more substantial, rising from 2.4% to 18.8%.
Gradical
Trare of Messin
Group
4..0
30.0
Proporivan
20.9
10.9
Takeaway 2: Class imbalance is to blame for unfairness, because of negative correlations between the group size and the gradient norm as well as the trace of Hessian.
0.
White
Black
Indian
Asian
Others
Figure 2. Proportions of gradient norms, traces of Hessian and group sizes for five demographic groups using full-po precision ResNe150 (fw-) ) on the UTK-Face dataset in ethnicity classifi- cation task.
5. Fairness in Quantization-Awar Training
This section extends the analysis the context of QAT. QAT models quantization during training and usually provides higher accuracy than PTQ schemes at lower bit-widths. The bound of excessive loss in QAT shares a large similarity to that in PTQ, but differs in two additional terms that are not negligible. To see details, let us consider one iteration + of updating parameters using widely-used optimizer, mini-bate SGD.
The effect of traces of the Hessian matrices. A simi- lar lemma establishes connection between the trace of a group's s Hessian and the size of the corresponding dataset.
Lemma 4.4. For given group Dap with a protected at- tribute Up E A {a1, .2..... ,0k), its trace of Hessian associated with the ERM function L evaluated at w² can be upper bounded by:
Recall that quantization stores the floating-pots parameter vector u as a fixed-point wint and computations are based on W. The update" is given by Krishnamoorthi (2018):
TH
Dap
1
(n|D) Amaz (mazz(H..) (HW-)
Dapl
B
(10)
Oe
(I (o:)).).
Daj Tr TT )i))
Wc+1 it
(11)
B
dw,
3FD9
where É assumed as a twice differentiable and convex loss function, 1/ is the learning rate, and B is the mini-batch. And w. is approximated by w, after the quantization and de- de-quantization process, which is given by:
where i E [kt] k: = |A| 2 2 and og is any attribute in A except Op.
Lemma 4.4 also indicates that the trace of the Hessian ma- trix for a specific group is related to the size of that group. Experiments in Figure 2 give evidence: groups with smaller sizes have larger traces of Hessians and vece-versa, i.e., wn' P2t) data points generally exhibit larger traces of Hessians com- pared to larger groups and vice-versa, resulting in more pronounced unfairness towards underrepresented groups.
W. = clammp( w 2 121,2-1111 -
(12)
Tr Tr(H
D.... This suggests that groups with fewer
where 8 the scale inversely proportional to bit-width b. Further, recall that this induces an error Aw; w. w. upper bounded by 1 Vnssaaz.
The following result sheds light the unfairness induced by QAT and provides a useful upper bound for a group's excessive loss.
Experiments on a balanced dataset. To further support our findings in Lemma 4.2 and Lemma 4.4, we experiment on balanced dataset CIFAR-10 (Krizhevsky et al., 2010)
Theorem 5.1. Let be twice differentiable loss function and consider a training process as defined in Eqn. (11) and quantization process as defined in Eqn. (12). Then, the excessive loss gap for group E A at iteration t + is
'We also experiment on another balanced dataset MNIST with similar findings. Besides, for a clear comparison, we set up experi- ments on artificial Imbalanced Ciw.eneeed---FR------ CIFAR- 10 and mabalanced-MNIST) whose training data is constructed by sampling different portions of images among different classes. Details and results in Ap- pendix C.I and Appendix C.2.
"The update relies on straight-through estimator (STE) (Ben- gio al., 2013), i.e. /rn 1. Details in the proof of Theorem 5.1.
6

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Understanding the Unfairness in Network Quantization
upper bounded by:
ent updates must adapt to quantization-indueed noise, lead- ing to optimization in a more distorted loss landscape. In regions with high Hessian traces, the steep loss surface am- plifies the effect of large gradient norms, causing uneven updates across subgroups. In contrast, PTQ quantizes only after full- pell-preciioon training, avoiding these interaction ef- fects and resulting in relatively lower unfairness.
1
1
Ga+i(a) S
vns
1 Da| || + 8
max Tr(H":
118
2
upper bound of (aa under PTQ
(13a)
+ V max 119 (1+ 3m Tr(HW:: ))
2
Takeaway 3: Although quantization-aware training al- ways provides a better overall performance guarantee, deterioration in fairness induced by imbalanced datasets towards protected attributes is much more severe than that in post-training quantization.
+ n15 man"ll Tr (H we 0
(13b)
where [9 and Tr H wi) are defined in the same way as in Theorem 4.1.
Although Eqn. (13a) and Eqn. (13b) give larger bound than that in Theorem 4.1, they also shed light relation between fairness and the gradient norm | and the trace
6. Mitigation Scheme and Evaluation
To further validate our main findings from theoretical and experimental analyses, that is, imbalanced datasets induce unfairness in model quantization, we adopt several data augmentation techniques as mitigation schemes. Data aug- mentation is viewed as a very powerful method to improve the generalizability of a deep model, especially in vision tasks, basically because the augmented data often provides more comprehensive representation of data points, thus minimizing the distance between the training and validation set. In addition, data augmentation is also investigated as an effective way to alleviate class imbalance. We refer readers to a survey by Leevy et al. (2018) for details.
of the Hessian Tr H :i in QAT. Similar theoretical con-
clusions can be carried over from Theorem 4. 1, Lemma 4.2,
and Lemma 4.4. We only present our empirical results here as a verification.
Figure 1(b) and Figure 2 show the relationship between the gradient norm, the trace of the Hessian matrix, and the model accuracy, for five demographic attributes and their corresponding subgroups, at three different bit-widths in QAT. Observations are quite similar to those from Fig- ure 1(a) and Figure 2: (1) For fixed bit-width, groups with smaller gradient norms and traces of Hessians tend to have higher accuracy and vice-versa; (2) The accuracy of each group decreases as the bit-width gets lower, however, the accuracy decreases faster for groups with larger gradient norms and traces of the Hessian matrices and vice-versa In addition, we also conduct experiment on CIFAR-I0 in Figure 4(b), in which unfairness in accuracy is not worsened significantly. As presented in Table I, the value of (DD only increases slightly, from 1.1% to 1.5%.
We consider two commonly used methods for image aug- mentations: geometric transformations and random erasing (Zhong et al., 2020):
Geometric Transformations (GT). For each class, randomly pick an image each time, and adopt one transformation from: (1) rotation with maximum degree of 20; (2) zooming with factor between 1.1 and 1.2; (3) horizontal flipping, until enough images are generated.
Comparison to PTQ: a larger gap in Eqn. (13b). Recall that Theorem 5.1 decomposes the excessive loss G:+1 (a) at ++ 1-th iteration into two key components: the first term (13a) is identical to that in PTQ in Theorem 4.1, and the extra term (13b) relates together with gradient norms and traces of Hessian, which gives a larger bound than PTQ. To provide clear comparison of this, we demonstrate in Table 1 on how fairness metric (DD changes as bit-widths get lower. As shown, DDD) in QAT grow significantly larger than PTQ, echoing the difference in the theoreti- cal bound. For instance, on the UTK-Face dataset with ResNet50, (DD rises from 2.4% at int 16 to 18.8% int 4 for PTQ. while for QAT, it increases from 3.3% to 25.4%. To provide more intuition, QAT exacerbates unfairness more than PTQ due to the dynamic interaction between gradient norms and Hessian traces under quantization constraints. Since QAT applies quantization throughout training, gradi-
Random Erasing (RE). For each class, randomly pick an image to be augmented each time. How to choose a mask of size n xm is crucial in the effectiveness of such method. To simplify the process, we adopt a random strategy for selecting the mask size by enforcing square mask, i.c., setting m. This is appropriate for datasets such as UTK-Face and FER2013, where images are uniformly sized at 48 x 48 pixels. During each augmentation step, a mask size parameter n is randomly sampled from the set {3, 4,.., .20). derived from the optimal configuration in (Zhong et al., 2020). A patch of size the image and masked with random values.
is then randomly selected from
We evaluate the effectiveness of both methods by training ResNet-50 and VGG19 on -TCe-Fae and FER2013. We
7

---

Understanding the Unfairness in Network Quantization
0.90
930
0.90
0.90
Mick
Blck bdie
Blcck
0.58
Ida
935
0.38
Tudaa
0.53
MBE
0.56
936
0.80
0.80
0.64
034
0.84
6.64
0.62
682
0.82
0.82
0.50
980
0.80
0.80
(a) Goometric Tramsfrmaaions PTQ
(b) Random Erasing in PTQ
(6) Geometric Transformations in QAT
Random Brasing QAT
Figure Mitigation schemes: Accuracy across groups using ResNe150 on UTK-Face dataset by PTQ ((a) and (b)) and QAT ((c)
(d)). Both GT and RE are adopted on for data augmentation.
Table 2. Fairness metric (DD for ResNet50 with data augmentation methods (GT and RF) applied.
(D) (%)
Augmentation Method
Quantization Method
UTK-
FER2013
int 16
int
int
int 16
int
int
PTQ
2.4
3.6
18.8
2.9
6.9
13.4
Non-Mitigation
QAT
3.3
6.0
25.4
3.6
9.9
44.4
PTQ
0.8
1.1
2.6
1.8
2.3
2.9
GT
QAT
1.3
1.7
2.7
2.1
2.5
3.2
PTQ
1.0
1.2
1.7
2.0
3.2
4.8
RE
QAT
1.1
1.3
2.0
2.8
3.9
4.9
present the results of training ResNet50 on UTK-Face using both PTQ and QAT, and leave other empirical evaluations to Appendix C.4. The UTK-Face dataset contains 18,964 images in training set, with 8, 108 for White, 3,611 for Black, 3,176 for Indian, 2,718 for Asian, and 1,351 for Others. We balance them by augmenting training images until each containing 8,108 images.
As shown, whether under PTQ or QAT, fixing n 3 or n = 20 is almost ineffective in mitigating unfairness. The fixed 10 strategy offers a slight improvement in fairness, but none of these fixed approaches are as effective as the random selection strategy for n. This suggests that, beyond the amount of augmented data, its quality also plays a crucial role in mitigating unfairness.
Figure presents the accuracy across five groups by PTQ (a) and (b)) and QAT ((c) and (d)). Both GT and RE are adopted for data augmentation. Comparing to Figure I, unfairness is significantly mitigated for both cases. For ex- ample, in QAT, the accuracy of group Others drops from 42% for 32-bit to 7% for 4-bit without data augmentation, while they drop from 82% and 83.2% for 32-bit to 80.3% and 82% for 4-bit when GT and RE are adopted, respec- tively. Furthermore, shown in Table 2, the values of yo( (D) significantly reduced after applying data augmentation methods, effectively mitigating unfairness.
Takeaway 4: Data augmentation helps mitigate the fairness caused by quantization.
7. Conclusion and Future Work
Starting from an observation in an experiment on quantiza- tion and accuracy across different subgroups of UTK-Face, we are aware significant concern for equality protection and privacy preservation. To the best of our knowledge, this is the first paper to both address this issue and go in-depth its causes and mitigation theoretically and empirically. Our main findings are concluded in "Takeaways". For fu- ture work, we will validate our findings in other vision tasks, such as detection, or other areas where compressed deep models are needed, such as NI.P. Besides, we aim to explore solutions beyond data augmentation mitigate unfairness.
Besides, as another evidence, our experiments on CIFAR-10 (and Imbalanced- CEa--CIIAR--1 and MNIST (and Imbalanced- MNIST) also support the effectiveness of data augmentation. See Appendix I and Appendix C.2 for details.
We also compare the above random selection strategy of RE to some fixed choices of n, including 3, 10 and
20. The corresponding results are presented in Table 3.
8

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Understanding the Unfairness in Network Quantization
Table Fairness metric yo( D) ResNet50 with fixed patch sizes n over range {3, 10, 20), compared with the random selection strategy of n from the range (3, ...,, 20) (referred to the Optimal Strategy) and data augmentation (referred to as the Baseline). in random crasing.
(D (%)
Quantization Method
Patch Size
UTK-Face
FER2013
int 16
int 8
int 4
int 16
int 8
int 4
PTQ
2.4
3.6
18.8
2.9
6.9
13.4
Baseline
QAT
3.3
6.0
25.4
3.6
9.9
44.4
PTQ
1.0
1.2
1.7
2.0
3.2
4.8
Optimal Strategy
QAT
1.1
1.3
2.0
2.8
3.9
4.9
PTQ
2.2
3.3
18.8
2.5
6.3
12.1
72 =
QAT
3.1
5.8
25.0
3.2
8.7
40.3
PTQ
1,4
2.1
10.6
2.3
4.3
8.7
n = 10
QAT
1.5
2.4
11.9
3.0
5.6
10.5
PTQ
2.3
3.6
18.7
2.7
6.5
12.8
20
QAT
3.3
5.9
25.2
3.3
9.1
41.9
Acknowledgements
3316-3333, 2021.
Qiankun Zhang is supported by the National Natural Science Foundation of China (Grant 62302183), Open Foundation of Key Laboratory of Cyberspace Security, Ministry Educa- tion China (Grant KLCC20240401), Ant Group Research Fund (Grant 20242452) and CCF-DiDi GAIA Collaborative Research Funds (Grant CCF-DiDi GAIA 202412). Jing Wang is supported by the National Natural Science Foun- dation of China (Grant 62202197) and the Major Research Project of Hubei Province (Grant 2023BAA027). Bin Yuan supported by the National Natural Science Foundation of China (Grant 62372191), the Open Topics from The Lion Rock Labs of Cyberspace Security (Grant LRL24013), and Songshan Laboratory (Grant 2411 102 (0210200). Xianjun Deng supported by the National Key R&D Program of China (Grant 28600,)0336000 and the National Natural Science Foundation of China (Grant U24B20153).
Badar, M., Sikdar, S., Nejdl, W., and Fisichella, M. Fair- trade: Achieving pareto-optimal trade-offs between bal- anced accuracy and fairness in federated learning. In Proceedings of the AAAI Conference on Artificial Intelli- 2024.
gence, volume 38, pp.
Bengio, Y., Léonard, N., and Courville, A. Estimating or
propagating gradients through stochastic neurons for con- ditional computation. arXiv preprint arXiv: 13V.:3308.33. 2013.
Blakeney, C., Huish, N., Yan, Y., and Zong, Z. Simon says: Evaluating and mitigating bias pruned neural networks with knowledge distillation. arXiv preprint 20Xiv:216.077880 2021.
Caton, S. and Haas, C. Fairness machine learning: survey. ACM Computing Surveys, 56:1-38, 2024.
Impact Statement
Choi, J.. Wang, Z., Venkataramani, S., Chuang, P. L.-.,, Srini- vasan, V., and Gopalakrishnan, K. Pact: Parameterized clipping activation for quantized neural networks. arXiv preprint 110iv:1805.00 06085, 2018.
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Deng, L. The mnist database of handwritten digit images for machine learning research [best of the web]. JEEE signal processing magazine, 29(6):141-142, 2012.
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Understanding the Unfairness in Network Quantization
A. Missing Proofs
A.1. Proof of Eqn. (4)
Proof. Let w = (W1, W2, W.,, ................................ wn, be an n--dimnnsional vector whose component w, a real number. w the approximate vector of ur obtained after us has been quantized and de-quantizzd.
Since the distribution of the parameters Wmin mar- selecting scaling factor 2--1-1 ensures that the quantization Eqn. (3) does not incur clipping errors. The reason is as follows:
[iii Wmar] of the neural network is usually symmetric about O, i.e.,
W maa
2
W -2 +1,
2 < 22 15
w < 2 - 1.
Thus, the quantization error of this method comes only from the [-] operator, i.c.:
Aw; = i wi S 8: 2 2
where s, in the inequality comes from the coefficient of the de-quantization formula Eqn. (2) <1), 28 comes from the error of the 1-] operator.
For the & norm of the error vector Aw:
||wl|| = (|A Aw2, ... Aw; ... Aw,)ll
1
Aw?
<
4
nsa
mar
2
v ns max
where Smaw is the largest scale used in the quantization among all parameters.
A.2. Proof of Theorem 4.1
Proof. First recall that we assume the loss function {(-) is twice differentiable. We use second-order Taylor expansion around w" The ERM function L(w"; Da) for group a E A can be stated as:
Llw;; Da) L(w"; Da) (T (Aw (Amn" ) (11Aw))1*)
The excessive loss G(a) for group a e Ais then given by definition:
((a) = L(wi; Da) (D)
1 1 ( H (Aw") o (1Aw0)))
(Aw") )T gh:
Further recall in Equ. (4), the quantizaton error |Aw"| = || w" is upper bounded by:
Aw" 2 Vnss
Combining with a Cauchy-Schwarz inequality, follows by:
(Aw") To < ||w""|| [g 1I
Vnismas
11
(14)
2
For the second-order derivative term, combining Eqn. (4) and the consistency between the matrix E, norm and the vector b norm, we have:
: ) (Aw") TH (Aw") 5 Al
|
S
ns
maz
H D.
8
12

---

Understanding the Unfairness in Network Quantization
Besides, since H D= is real symmetric matrix, | max; Aal, where X the eigenvalue of H Moreover, the matrix = is positive semi-definite, meaning all eigenvalues of the matrix = greater than equal D. This local
property holds for convex loss functions and also for non-convex ones, the second-order Taylor expansion optimum approximates the loss as convex (Nocedal & Wright, 1999). So H max; XІ. According to the property of the trace of the matrix that the trace a matrix is equal to the sum of all eigenvalues of the matrix, we have:
H - max A S Σ = =x
Thus,
(Aw")TH (Aw") ns maz ()(
(15)
The upper bound for the excessive loss (a) is thus obtained by combining Eqn. (14) and Eqn. (15).
A.3. Proof of Lemma 4.2
Proof. By the assumption that the model converges to local minimum, it follows that:
g
V w L(w" D)
(m)
9 ww"
D
Daj
(2))
gw-
D
j=1
(16)
Dayl
Day
(w).))
gw-
|aag
Dayl Day D Iw
-0.
Thus, for group a; E A, we have:
9w-
Da.
Σ
Daylow
D
Besides, by the trigonometric inequality property of the vector norm, it follows that:
k Dayl
De,
1
-11gm"
Day
||
(17)
Da.
Add I || to both sides of Eqn.( 17):
k
2
Day
1
1
Dajl
119..
Day
(18)
D
jml
Divide both sides of Eqn. (18) by 2:
||g m0
Da:
5
D ay
D |19 1m
2 2
0.
j=1
13

---

Understanding the Unfairness in Network Quantization
A.4. Proof of Corollary 4.3
Proof. For groups and b in A, by the assumption that the model converges local minimum, it follows that:
gw- Ww
[ww" D)
D
Σ
(m::Ds)
D
9
w
(m)
Iw-
9 9w
D
D
D.
Da
Da
(mi r
Dol D
(a))
9 w
D
D
Do
i=1
Da
Do
D
D.
D
9
w
|D]
9w
0.
Thus, we have:
D
D Da
9
9 ww
| -
Da
Dal S [Dol then ||
| Da
||-
A.5. Proof of Lemma 4.4
Proof. In the same way as the derivation of p in Eqn. (16), we have:
H
Σ
k
D
H
Day
D
By the simple property of the trace of the matrix and the relationship between the trace and eigenvalues of the matrix, it follows that:
Tr(H Tr
D
Dayl
Tr H w-
Duj)
S admma (H DD
D
For group ap. we have:
Dapl D
Tr (H(H DuP) S ndmaz (HD.)
D aj
Tr(H P:y
Tr
D
j&p
Day
S ndmaz (HD.) )
Tr H Bay w0
Thus,
k Σ
Day Tr H
11 D/maz HDD.)
Day Tr (H Day
10
Dapl
itpa
A.6. Proof of Theorem 5.1
Proof. The proof of Theorem 5.1 relies on the following two second order Taylor approximations: (1) The first approximates the ERM function iteration & under non-quantized training, i.e., u1+1 wr - B. where B D denotes the
14

---

Understanding the Unfairness in Network Quantization
mini-batch. (2) The second approximates the ERM function under quantized training, where the parameter vector is updated
by:
B
ae
(sū (3::), U1)
u0++1
iwe
13
Oe
Nw.
(Si (2:4),
u7
B
Ori
Ow
11
B
0l
(fm, (2x;) 31:
w
dwi
799 w.
Chi 1 by the straight-through estimator (STE) (Bengio al., 2013; Nagel et al., 2021). Finally, the result is where Pan.
obtained by taking the difference of these approximations under quantized and non-quantized training.
Approximation non-quantized ERM. The approximation of non-quantized ERM can derived using a second order Taylor approximation as follows:
gw: f(w++1; Da) L(w, ngw Da) C(w;; Do) ((gg.)" TT. " "
1 T gw:
(19)
2. Approximation of quantized ERM. Consider one iteration + 1 of updating parameters using the mini-batch SGD, i.e. Wt+1 = w. Tg Recall that quantization stores the float-point parameter vector w as a fixed-point Wint and computation is based on w. w1+1 approximated by w when participating in the next round of iterations of the computation. Hence,
W = W++1 + Aw1+1 = w. - ng - +
Awi+1 = w; + Aw;
B 179 w
+ Aw
Applying a second order Taylor approximation around w, allows us estimate the quantized ERM function at iteration 1 as:
[(we+1: Da)
=L(w, + Aw, + Aw:+1 - :1 Da)
(20a)
~(ww;; Da) + (Am, Am, 119 ". YT Iw:
D.
(20b)
+ (Aw; + Awi+1 "gg.. B YT H D. (wu + Aur+1 -19 m9
D(w; Da) + (Am, + Am 1 119 w.
H B wi
Aw,) gw.
(20c)
+ (Aw; Awi+1 ng B, nH 81 Aw;)
"H D., (Awi + Aw ng
nH B Awi)
= -[(w; Da) + (Aw; + 1 gw: 77 w.
D.
T H w. Aw;) 9w:
(20d)
Aw, Aw T
T (Aw; Aw
1 /"
(g
"HH Dz gw.
- (n,; + Aw wi
Da D v/Aw; + Awo ) H HD HB Aw,) + O(1²),
wi
where Eqn. (20c) follows from the first-order Taylor expansion approximation of 인 at w.:
B wn
~ gw. +
Aw,HD.
The upper bound for the excessive loss (a) can be obtained by combining Eqn. (19) and Eqn. (20d):
15

---

Understanding the Unfairness in Network Quantization
((+1(a)
=f(ii+1: D.) L(W1+11; Da)
(21a)
=(Aw, + Aw(+1)
" ww;
D. ( w, Aw;) Iw: " + 2 (Aaw; + " P= (Aw, + Aw)+1)
(21b)
(nw + Aw
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The Eqn. (21d) follows from that the mini-bater B is randomly selected from Da. the average gradient norms and average Hessian matrices in B are approximate to those in D., And Eqn. (21e) follows from the upper bound of | and the relationship between the trace and eigenvalues of the matrix.
B. Experimental Settings
B.1. Datasets
Table Datasets used in our experiments.
Dataset
Description
Training Set
Test Set
Labels
UTK-FFace
Face image
18,964
4,741
Age, gender, ethnicity
FER2013
Facial expression image
28,708
7,178
Seven facial expressions
CIFAR-10
RGB image
50,000
10,000
Ten object classes
Imbalanced-CIFARR110
RGB image
19,375
10,000
Ten object classes
MNIST
Handwritten-dight image
60,000
10,000
0-9
Imbalanced-MNIST
Handwritten-digits image
23,782
10,000
0--9
Table 4 shows all datasets used this paper. We experiment on the imbalanced UTK-face and FER2013 and the balanced CIFAR-10 10 and MNIST. Besides we artificially construct two imbalanced datasets -IFARR------R-1O0 and Imbalanced- MNIST by randomly and proportionally discarding images from each class. Details for usage of each dataset presented in the corresponding subsection in Appendix C.
B.2. Models and Training Details
The paper adopts the following models verify the results of the main paper:
ResNet18 (He et al., 2016). This model consists 17 convolution layers, AvgPool layer and 1 fully connected layer and has 11.7 million parameters.
16

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Understanding the Unfairness in Network Quantization
ResNet50 (He al., 2016). This model contains 50 convolution layers, AvgPool layer and fully connected layer and has 25 million parameters.
VGG19 (Simonyan & Zisserman, 2014). This model consists of 19 layers (16 convolution layers, fully connected layers,5 MaxPool layers and SoftMax layer) with 143 million parameters.
The datasets and corresponding models used for all experiments in this paper are as follows:
Experiments on the UTK-Face dataset using the ResNet50 and VGG19 models.
Experiments on the CIFAR-10 and MNIST datasets using the ResNet18 model.
Experiments the FER2013 dataset using the ResNet50 and VGG19 models.
Experiments on the Mubalanced-CIFARR and Imbalanced- MNIIST datasets using the ResNet18 model.
The training process is performed using NVIDIA 3090Ti device. The hyperparameters for all the models are set with an initial learning rate of 0.001, which is gradually reduced based the number of epochs during training to optimize the models. The VGG19 model is trained for epochs ranging from 40 to 60, the ResNet18 model also undergoes training for 100 epochs, while the ResNet50 model is trained for approximately 200 epochs.
All experiments are conducted in a Python 3.10 environment using the PyTorch framework. For each experiment, the overall classification accuracy is calculated, along with the classification accuracy on different protected group sets, as well as the gradient norms and the traces of the Hessian matrices. Each experiment is repeated ten times to calculate the average results.
C. A Additional Experimental Results
C.1. Experiments on (balanced) CIFAR-10 and MNIST using ResNet18
In this section, we conduct experiments on ResNet18 8 for two balanced datasets CIFAR-10 (Krizhevsky et al., 2010) and MNIST (Deng, 2012), and further support our findings in the main paper by observing the faimess phenomena in balanced datasets. Figure 4 reports our results and shows that as the bit-width gets lower in PTQ and QAT, accuracy gaps between 10 groups on both CIFAR-10 10 and MNIST is relatively stable, while they grow larger on UTK-Face in Figure 1. Furthermore, as summarized in Table 1, the faimess metric yo( for the datasets remains quite small across all bit-widths, indicating the model performs more fairly on these datasets compared to others with higher fairness metric values.
0/6
Aceurac
0.59
67-
001
4.4
08
Cal
092
0.52
Hore
0.5
056
0.50
(4) CIFAR---O under PTQ
(1) CIFAR-19 under QAT
Mc)NT under PTQ
(40 MINSST under QAT
Figure 4. Experiments on CIFAR-10 10 and MNIST datasets on the accuracy of each subgroup of individuals using ResNet18. Both PTQ and QAT evaluated as bit-widths get lower.
C.2. Experiments on Imbalanced-
and Imbalanced-MNIIST
To further validate the impact of imbalanced datasets in quantization the faimess of the model, manually construct two imbalanced datasets, Imbalanced-CIFia 10 and Imbalanced-MNIIS" datasets, based on CIFAR-10 and MNIST datasets, respectively, and experiment on the ResNet50 model.
We construct the Miballaned-CIFAR... in the ratio of 16: 16: 8: 4: 4 2: 2:: 1: from the 10 object categories of Plane, Car, Bird, Cat, Deer, Dog, Frog, Horse, Ship and Truck, respectively. Similarly, we construct the Imbalanced-MNN dataset by randomly selecting a total of
dataset by randomly selecting a total of 19375 images from the CIFAR-10 dataset
17

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Understanding the Unfairness in Network Quantization
23250 images from the MNIST dataset in the ratio of 16 16 :6:8::::::: categories of "0", "1", "2", "3", "4", "5", "6", "7", "8" and "9", respectively.
8 8: 4
4:2: 2:: from the 10 handwritten digit
Gratical Norm
Trace Bosian
0.3
0.-
Group Size
20.0
Acairacy
0.6
Accuracy
0.6
0.4
Plane
Dog
0.4
Dog
Car
Freg
Car
Frog
0.3
Bird
Horst
0.2
Bird
Horse
Cat
Ship
Sháp
5.8
Dear
Truck
Duzr
Truck
0.0
0.8
Doat 32
Int
iut
iuc
float 32
int 8
Car
Blrd
Cut
Deer
Frog Ship Truck
(a) PTQ
(b) QAT
(c) Group sizes vs. Gradient norms vx. Traces of Hessiuns
Figure 5. (a) and (b) represent the experiments on CIFAR-I0 dataset on the accuracy of each subgroup of individuals using ResNet18 for a gender classification task: (c) represents the proportions of gradient norms, traces of Hessian and group sizes for four demographic groups.
1.9
1.0
30.0
Gradieot Norn
Trace
10
Group Size
0.9
0.9
Accuracy
Accuracy
0.9
0.9
15.
0
5
5
0.8
7
0.5
0.7
0.7
0.8
Boott 32
Int 16
Int 5
int
Doot 32
lat 16
Int 8
lut
Plane Car Bird
Cat
Deer Dog
Frog Ship Truck
(a) PTQ
(b) QAT
(c) Group size Gradient norms Traces of Hessians
Figure 6. (a) and (h) represent the experiments on MNIST dataset on the accuracy of each subgroup of individuals using ResNet18 for gender classification task; (c) represents the proportions of gradient norms, traces of Hessian and group sizes for four demographic groups.
The experiments in Figure 5 and Figure 6 show the relationships between the gradient norm, the trace of the Hessian matrix, and the model accuracy at three different bit-widths in PTQ and QAT, with Figure 5 corresponding to Jabalanced-CIFAA 10 and Figure 6 to Imbalanced-MNNIST. The experimental results also show very similar trends to those reported in the main body of the paper. Taking Figure as an example, groups Frog, Horse, Ship and Truck with larger gradient norms and traces of Hessians compared to the other groups, the accuracy drops off faster. And for a specific bit-width, for example int 4, the accuracy is lower these groups. Consequently, as the quantization bit-width b decreases, the disparity in ((a) values across groups becomes more pronounced, leading to an increase in (D) and further exacerbating unfairness.
C.3. Experiments on different models using different imbalanced datasets
Gradian Norm
Trace of Hassian
0.7
40.0
Group
Aecuracy
Progartion (5)
30.0
0.5
0.5
0.4
0.4
0.3
White
0.3
White
Black
0.2
Indian
0.2
Indian
80.0
Asian
Asisn
0.1
Others
0.1
Others
6.0
0.0
16
White
Black
Asian
Orhers
(a) PTQ
(b) QAT
(c) Group sizes vs. Gradient norms vs. Traces of Hessians
Figure 7. (a) and (b) represent the experiments on UTK-Face dataset on the accuracy of cach subgroup of individuals using VGG19 for a gender classification task; (c) represents the proportions of gradient norms, traces of Hessian and group sizes for four demographic groups
18

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Understanding the Unfairness in Network Quantization
To further validate the theoretical and experimental analyses in the main paper, we conduct experiments on different datasets and models to evaluate the impact of quantization on model faimess. In addition to the experiments the UTK-Face dataset on the ResNet50 model in the main paper, we additionally conduct experiments on the UTK-Face dataset using VGG19 model, and the FER2013 dataset using ResNet50 and VGG19 models.
For the UTK-Face dataset, we set up an ethnicity classification task on the VGG19 model with a protected group that coincides with the target label, i.e. White, Black, Indian, Asian and Others.
For the FER2013((oodiee) VGG19 models with a protected group set that coincides with the target label set, i.e., Happy, Sad, Neutral, Fear, Angry, Surprise and Disgust.
al., 2013) dataset, we performed facial expressions classification tasks on the ResNet50 and
0.9-
360
Geadiout Nermn
0.5
Trace of Blddan
0.7
25.
Gevup Size
Accuraty
0.6
Proportan (55
20-0
0.5
15.0
0.4
Heppy Nestrel
Ange
Hagpp
Angry
ourpeoo
Dirgutt
float 32
16
Dout
int
Hagpy Neutral
Sad
Fear
Angry SurpriseDiguut
(a) PTQ
(b) QAT
(c) Group sizes vs. Gradient norms vs. Traces of Hessians
Figure& 8. (a) and (b) represent the experiments on FER2013 on the accuracy of each subgroup of individuals using ResNet50 for gender classification task; (c) represents the proportions of gradient norms, traces of Hessian and group sizes for four demographic groups.
0.9
0.9
30.
Gradirnt Norm
0.8
0.8
Hestion
0.7
0.7
25.0
Group Siwe
0.6
0.6
200
Accuracy
0.5
Accuracy
0.5
Propartion
25B
0.4
0.4
0.3
0.3
10.8
0.2
Baypr
0.2
Happy
Neatral
Suguie
Nearal
Surprixe
5.0
0.1
Sad
0.1
Sed
Digui
Fox
0.0
0.0
0.5
Bost 32
int 16
int 4
Thoot 32
int 16
int 8
int
Happy Neutral
Smd
Fear
Angry SarpriseDiggss
(a) PTQ
QAT
(c) Group sizes
Gradient norms
Traces of Hessians
Figure 9. (a) and (b) represent the experiments on 2013 dataset on the accuracy each subgroup of individuals using VGG19 for a gender classification task; (c) represents the proportions of gradient norms, traces of Hessian and group sizes for four demographic groups.
The experiments in Figures 7-9 show the relationships between the gradient norm, the trace of the Hessian matrix, and the model accuracy at three different bit-widths in PTQ and QAT. The experimental results show very similar trends to those reported in the main body of the paper: (1) Consider a specific bit-width, for example, int 8 in Figure 7. While the gradient norms and the traces of the Hessian matrices are larger over different groups, e.g., comparing group White and Others, there is an opposite numerical relationship in accuracy: (2) Consider specific demographic group, for example, group Others in Figure 7. While its gradient norm accounts for the largest percentage, its accuracy decreases the most fast. Similar phenomena happen in traces of the Hessian matrices. As consequence, the reduction in quantization bit-width b causes the gap in (a) values among different groups to widen further, resulting in higher (D) and thereby intensifying unfairness. These results align with those presented in Table 1.
In a similar manner, to clearly compare the unfairness in PTQ and QAT, Table 1 demonstrates how the fairness metric sp (D) evolves as the bit-width decreases. As evident, (DD) in QAT increases substantially more than in PTQ
19

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Understanding the Unfairness in Network Quantization
C.4. Supplementary experiments mitigation schemes
Evaluating our mitigation scheme on different (imbalanced) datasets using different models. To further illustrate the effectiveness of GT and RE in mitigating unfairesss, We also conduct experiments using ResNet50 and VGG19 on the (imbalanced) FER2013 for both PTQ and QAT as follows:
Experiments for an ethnicity classification task with ethnicities as the protected attributes on the UTK-Face dataset and VGG19 model. The specific augmentation methods for the five ethnicity groups in UTK-Face are consistent with those described in the main paper.
Experiments on the ResNet50 and VGG19 models using the FER2013 dataset. The FER2013 dataset consists of 28,700 708 training images, distributed as follows: 7,091 images for Happy, 4,966 for Neutral, 4,937 for Sad, 4,162 for Fear, 3,818 for Angry, 3,301 for Surprise and 431 for Disgust. To address this imbalance, we augment the training images so that each age group contains 7,091 images.
As shown in Table 5, the values of (D) are considerably reduced, highlighting the effectiveness of data augmentation in mitigating unfairness.
Table 5. Fairness metric (D) for VGG19 with data augmentation methods (GT and RE) applied.
(D( (%)
Augmentation Method
Quantization Method
UTK-Face
FER2013
int 16
int
int
int 16
int
int
PTQ
11.2
12.4
54.6
0.6
1.5
50.7
Non-Mitigation
QAT
11.7
18.9
84.1
1.4
2.9
65.7
PTQ
0.4
1.0
1.9
1.4
1.7
2.5
GT
QAT
1.2
1.5
2.4
1.4
1.8
3.0
PTQ
0.5
1.2
1.5
1.7
2.0
3.7
RE
QAT
1.0
1.3
2.0
1.8
2.2
4.1
20


Reviews:
[{"review_id": "ar2YIHWMtp", "reviewer": "Reviewer_Tpv5", "summary": "The paper investigates the fairness implications of network quantization, focusing on two widely used algorithms: Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). \n\nThe authors identify two key factors that exacerbate unfairness in model accuracy across different groups: the gradient norm of the group loss function and the trace of the group loss function's Hessian matrix. They theoretically analyze and empirically validate these factors, showing that class imbalance leads to distinct values of these factors among different attribute classes, which in turn exacerbates unfairness. \n\nThe paper also compares PTQ and QAT, finding that QAT, while generally preserving higher accuracy at lower bit-widths, exacerbates unfairness more severely than PTQ. To mitigate this unfairness, the authors propose and evaluate simple data augmentation techniques, demonstrating their effectiveness in reducing disparate impacts of quantization.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your acknowledgment that “the paper addresses an important and timely issue, the theoretical analysis is rigorous, and the empirical evaluation is thorough.” We believe that our experimental results strongly support our theoretical findings. ​In response to your concerns, we have conducted further experiments and confirmed the consistent effectiveness of our method. The detailed results are available at https://anonymous.4open.science/api/repo/Rebuttal-30C1/file/Reviewer%20Tpv5.pdf?v=917ec334. \n- W1: ​Our intention to conduct experiments on Imbalanced-CIFAR-10 and Imbalanced-MNIST is to facilitate **direct** comparisons with CIFAR-10 and MNIST, highlighting the impact of class imbalance on fairness. In addition, we also evaluated our method on UTK-Face and FER2013, two naturally imbalanced datasets. To further address your concern, we have included new experiments on iNaturalist 2017, a real-world highly imbalanced dataset, in the revised version. Our findings remain consistent across all datasets (see Figures 1-2 and Table 1 in the link), reinforcing the real-world generalizability of our conclusions. \n- W2: We claim that our method is applicable to **any quantization precision**, including mixed-precision quantization (MPQ), as demonstrated by our quantization error analysis. While MPQ assigns different bit-widths to different layers or channels, the quantization error at each layer follows the same statistical properties as in uniform-precision quantization. To validate this, we have conducted experiments with MPQ on different models and datasets in the appendix (see Table 2 in the link). We have also added a discussion in Section 3.1 to clarify this point.\n- W3: We confirm that other strategies may also help address unfairness, and we have conducted experiments on reweighting and adversarial training (see Tables 3-4 in the link). Our results show that these methods can indeed help mitigate unfairness. We have expanded the discussion in the appendix. Regarding our choice of data augmentation as a mitigation strategy, our motivation is to provide a **more intuitive validation** of our theoretical conclusion that the exacerbation of unfairness in quantized models is related to class imbalance. To achieve this, we adopted geometric transformation and random erasing at the data processing stage, as these methods have been shown to be both effective in prior research and computationally efficient. In addition, we also explored training-stage strategies by introducing a regularization term in the loss function to penalize differences in gradient norms and Hessian traces across classes. However, this approach is less effective and more costly than data augmentation.\n- Q1: Our study specifically focuses on gradient norms and Hessian traces, both theoretically and experimentally, as they **directly impact** optimization stability and sensitivity to quantization errors. In addition, fairness in quantized models can be influenced by multiple factors. To further investigate this, we have included a new ablation study in the appendix that explores additional factors influencing fairness in quantized models. The results indicate that model architecture, optimization algorithm, and hardware selection all have potential impacts on fairness (see Tables 5-7 in the link). Specifically, the VGG architecture, Mini-batch SGD, and Ada L4 GPU tend to exacerbate unfairness in quantized models. \n- Q2: We believe that our fairness measurement method can be **directly generalized** to generative tasks, as it relies on the loss difference before and after quantization, with the cross-entropy loss in classification replaced by the appropriate loss function for generative models. To validate this, we have conducted experiments on VAE in the revised paper, using Evidence Lower Bound (ELBO) loss as the evaluation metric (see Table 1 in the link). Our results indicate that the fairness trends observed in classification tasks remain consistent in generative models, further supporting the generalizability of our findings.\n- Q3: Thank you for your valuable suggestion. To provide more intuition, QAT exacerbates unfairness more than PTQ due to the dynamic interaction between gradient norms and Hessian traces under quantization constraints. Since QAT applies quantization throughout training, gradient updates must adapt to quantization-induced noise, leading to optimization in a more distorted loss landscape. In regions with high Hessian traces, the steep loss surface amplifies the effect of large gradient norms, causing uneven updates across subgroups. In contrast, PTQ quantizes only after full-precision training, avoiding these interaction effects and resulting in relatively lower unfairness. We have added a more detailed discussion in the revised paper and believe this addition improves the accessibility of our theoretical insights.\n\nWe greatly appreciate your thorough review and thank you for reconsidering our paper.", "claims_and_evidence": "The claims made in the paper are generally well-supported by both theoretical analysis and empirical evidence. The authors provide a detailed theoretical framework to explain how gradient norms and Hessian traces contribute to unfairness in quantized models. They also conduct extensive experiments on multiple datasets (UTK-Face, FER2013, CIFAR-10, and MNIST) and models (ResNet and VGG) to validate their findings. The empirical results align well with the theoretical predictions, showing that groups with smaller datasets experience larger gradient norms and Hessian traces, leading to greater accuracy degradation after quantization.\n\nHowever, one potential issue is the reliance on synthetic imbalanced datasets (Imbalanced-CIFAR-10 and Imbalanced-MNIST) to validate the impact of class imbalance. While these datasets help illustrate the theoretical points, their artificial nature may limit the generalizability of the findings to real-world scenarios. The authors could strengthen their claims by including more naturally imbalanced datasets (e.g., iNaturalist).", "methods_and_evaluation_criteria": "The methods proposed in the paper are appropriate for the problem at hand. The authors use standard quantization techniques (PTQ and QAT) and evaluate their impact on fairness using well-established fairness metrics. The choice of datasets (UTK-Face, FER2013, CIFAR-10, and MNIST) is reasonable, as they cover both imbalanced and balanced scenarios, allowing the authors to demonstrate the impact of class imbalance on fairness.\n\nThe evaluation criteria, particularly the fairness metrics are well-defined and appropriate for measuring the disparate impacts of quantization across different groups. The authors also provide a clear explanation of how this metric is derived and why it is suitable for their analysis.", "theoretical_claims": "The theoretical claims in the paper are well-formulated and supported by rigorous proofs. The authors provide detailed derivations for the upper bounds of excessive loss in both PTQ and QAT, and they clearly explain how these bounds relate to the gradient norms and Hessian traces. The proofs are presented in the appendix and appear to be correct, though I did not verify every step in detail.\n\nOne minor point is that the authors could provide more intuition or discussion around the theoretical results, particularly for readers who may not be familiar with the mathematical details. For example, explaining why the interaction terms in QAT lead to greater unfairness compared to PTQ could help make the theoretical insights more accessible.", "experimental_designs_or_analyses": "The experimental design is sound and well-executed. The authors conduct experiments on multiple datasets and models, covering both imbalanced and balanced scenarios. They also perform ablation studies to validate the effectiveness of data augmentation techniques in mitigating unfairness. The results are presented clearly, with appropriate visualizations (e.g., accuracy plots, fairness metric tables) to support the findings.\n\nOne potential improvement would be to include more detailed ablation studies on the data augmentation techniques. For example, the authors could explore different augmentation strategies or hyperparameters to see how they affect the fairness of the quantized models. Additionally, the authors could provide more insights into why certain augmentation techniques (e.g., geometric transformations vs. random erasing) perform better in specific scenarios.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "**Strengths:**\n\n1. The paper addresses an important and timely issue in machine learning, namely the fairness implications of model compression techniques like quantization.\n\n2. The theoretical analysis is rigorous and provides clear insights into the factors that contribute to unfairness in quantized models.\n\n3. The empirical evaluation is thorough, covering multiple datasets, models, and quantization methods.\n\n4. The proposed data augmentation techniques are simple yet effective, and the authors provide clear evidence of their impact on fairness.\n\n**Weaknesses:**\n\n1. The reliance on synthetic imbalanced datasets (Imbalanced-CIFAR-10 and Imbalanced-MNIST)  may limit the generalizability of the findings to real-world scenarios.\n\n2. While the authors compare PTQ and QAT, they do not explore other quantization techniques (e.g., mixed-precision quantization) that might have different fairness implications.\n\n3. The authors propose data augmentation techniques to mitigate unfairness, but they do not explore other potential mitigation strategies (e.g., reweighting, adversarial training).", "other_comments_or_suggestions": "Please refer to the above comments.", "questions_for_authors": "1. The paper identifies gradient norms and Hessian traces as key factors contributing to unfairness in quantized models. However, are there other potential factors (e.g., model architecture, optimization algorithms) that could also influence fairness in quantized models?\n\n2. The paper focuses on fairness in the context of classification tasks. Have the authors considered whether their findings might extend to other types of tasks, such as generative models? \n\n3. The authors mention that QAT exacerbates unfairness more severely than PTQ due to the interaction between gradient norms and Hessian traces. Could the authors provide more intuition or a simplified explanation for why this interaction leads to greater unfairness in QAT compared to PTQ?", "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": "", "extra_scores": {}}, {"review_id": "VU0637CySP", "reviewer": "Reviewer_c7bP", "summary": "Network quantization, a widely studied model compression method, effectively converts floating-point models to fixed-point models with negligible accuracy loss. Despite its success in reducing model size, it can exacerbate fairness issues across different dataset groups. This paper examines Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT), identifying two primary factors causing these fairness issues through theoretical analysis and empirical verification. The study reveals that while QAT maintains higher accuracy at lower bit-widths, it performs worse than PTQ in terms of fairness. Additionally, simple data augmentation methods can mitigate these fairness issues, especially in cases of class imbalance. Experiments on imbalanced datasets (UTK-Face, FER2013) and balanced datasets (CIFAR-10, MNIST) using ResNet and VGG models validate these findings.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you sincerely for commenting that \"the detailed analysis and experiments provided for this issue are very convincing.\" We also truly appreciate your constructive suggestions. We have conducted additional experiments and provided further explanations to address your concerns. The additional experimental results are available at https://anonymous.4open.science/api/repo/Rebuttal-30C1/file/Reviewer%20c7bP.pdf?v=e20ea28c.\n- W1&Q1: Our theoretical analysis follows general PTQ and QAT principles and is designed to be applicable to various quantization methods. Specifically, in the final step of the derivation in Theorems 4.1 and 5.1, we only need to replace $\\Vert \\Delta w \\Vert$ with the corresponding quantization error of the specific method—for instance, when applying BRECQ[1], one would replace $\\Vert \\Delta w \\Vert$ with the Frobenius norm of the difference between the full-precision weights and the quantized weights. These theorems also indicate that the larger the quantization error of a method, the more severe the resulting unfairness. To further address your concerns, we have conducted additional experiments utilizing BRECQ for PTQ and LSQ[2] for QAT in the appendix of the revised manuscript, and the results are consistent with our original findings (see Table 1 in the link).\n- W2&W3&Suggestions: Thanks for your constructive suggestion. Actually, before submitting our manuscript, we faced a dilemma regarding the arrangement of the experiments between the main body and the appendix. In our previous submissions, we primarily focused on investigating the relationship between numerical characteristics of datasets (i.e., gradient norms and Hessian traces) and the fairness of quantized models through case-by-case analyses across different datasets, as you suggested. Moreover, we proposed mitigation strategies by introducing a regularization term in the loss function to penalize differences in gradient norms and Hessian traces across classes. \n\n  However, several previous reviewers questioned the practicality of the extensive discussion on Theorems 4.1 and 5.1. They argued that focusing on the numerical characteristics of datasets lacked intuitive and in-depth insights. Specifically, they pointed out that it was unclear what factors directly influence the gradient norms and Hessian traces, and they suggested that more theoretical and experimental investigation was needed. Additionally, they noted that the mitigation strategies based on gradient norms and Hessian traces, although effective, were impractical for real-world applications due to the high cost. \n\n  To address these concerns, we added further theoretical and experimental analysis of gradient norms and Hessian traces in the main body. We found that class imbalance significantly impacts gradient norms and Hessian traces. Consequently, we replaced the unfairness mitigation strategy with a simpler and more efficient data augmentation approach. These changes from our previous submission led to the current paper. \n\n  We greatly appreciate your suggestion to reconsider the arrangement of the content. In the revised paper, to further strengthen the validity of Theorems 4.1 and 5.1, we have moved the experimental results from Appendix C.4, which demonstrate the validity of these theorems on different datasets, to the main body. Meanwhile, we retained Lemma 4.2, Corollary 4.3, and Lemma 4.4, but relocated their theoretical analysis and experimental verification to the appendix for better clarity and focus.\n- Q2: We would like to clarify that our theoretical analysis for weight-only quantization (WOQ) can be directly extended to weight-activation quantization (WAQ). This extension is supported by the findings in [3], specifically Theorem 1, which shows that the impact of activation quantization on the target loss can be transformed into a similar effect as weight quantization. Based on this, we have derived the theoretical bounds for WAQ, which reveal that the upper bound of the excessive loss $G(a)$ introduces two additional terms compared to WOQ: $\\sqrt{n}\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}\\cdot\\Vert g_{w^*}\\Vert+\\frac{1}{2}n\\left(\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}\\right)^2\\cdot \\text{Tr}(H_{w^*})$. The coefficient $\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}$ has a stronger impact on fairness compared to the WOQ case. To further validate this theoretical analysis, we conduct experiments with WAQ and the results are consistent with our theoretical findings (see Table 2 in the link). We have included these theoretical analyses, proofs, and the corresponding experimental results and discussions in the revised paper.\n\nYour insights are valuable to us, and we appreciate your further surpport a lot.\n\n[1] BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction, ICLR 2021.\n\n[2] Learned Step Size Quantization, ICLR 2020.\n\n[3] QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization, ICLR 2022.", "claims_and_evidence": "The paper supports its claims through both experimental and theoretical evidence. Both the proofs and experiments effectively fulfill the role of supporting the claims made in the paper. However, some key experiments are included in the appendix.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense for the problem", "theoretical_claims": "I checked all proof. The proof is simple and clear.", "experimental_designs_or_analyses": "I think the experiments in appendix should be in main body of paper.", "relation_to_broader_scientific_literature": "This is a new topic", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "The paper offers innovative perspectives and analysis on quantization, particularly addressing the impact of different quantization methods on various classes, which is a novel topic. The detailed analysis and experiments provided for this issue are very convincing. However, the paper has several shortcomings:\n\n1. It lacks further analysis of different methods for Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT).\n2. I believe that the experiments in the appendix should be included in the main body of the paper, while some of the current main-body experiments should be moved to the appendix.\n3. The conclusion of the paper should focus more on providing theorems rather than analyzing data distribution, as datasets can vary significantly in their distributions. Judgments based on whether a single dataset is balanced or not are not sufficient.", "other_comments_or_suggestions": "The content from the quantization to Theorems 4.1 and 5.1 in this paper is excellent, elaborating on the mathematical principles behind the unfairness caused by quantization. However, I find the subsequent analysis of the relationship between the numerical characteristics of datasets and quantization inappropriate. Therefore, I suggest that the author analyze the numerical characteristics of datasets in relation to quantization case by case on different datasets. In the experiments, move the experiments from the appendix to the main body, focusing mainly on the validity of 4.1 and 5.1.", "questions_for_authors": "1.Even for the same Post-Training Quantization (PTQ), there are currently many quantization methods. It is unclear whether different quantization methods would affect the results of this paper. If possible, please provide proofs and experiments to demonstrate this.\n2. For model quantization, besides the quantization of parameters, there is also the quantization of activations. It would be helpful to provide an analysis of how the quantization of activations impacts the results.", "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": "", "extra_scores": {}}, {"review_id": "z8AXCPLr6i", "reviewer": "Reviewer_4FiB", "summary": "They used data enhancement to mitigate the unfairness of quantification of unbalanced data set models", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Great thanks for kindly evaluating that \"the content of this article is very interesting.\" We also sincerely appreciate your valuable feedback, and believe that the additional experiments, and explanations can address your concerns. The additional experimental results are available at https://anonymous.4open.science/api/repo/Rebuttal-30C1/file/Reviewer%204FiB.pdf?v=27c3e413.\n- Q1: We would like to clarify that our work goes beyond merely stating that \"class imbalance causes unfairness\" by offering detailed theoretical insights and empirical evidence to deepen the understanding of how class imbalance interacts with quantized model fairness, ultimately guiding the development of effective mitigation strategies. To enhance the clarity and impact of our conclusions, we have revised the manuscript to highlight the novel aspects of our analysis, such as the specific mechanisms by which quantization exacerbates unfairness in imbalanced datasets and the effectiveness of proposed mitigation strategies. To further address your concerns, we have included a new ablation study section after Section 5 to explore potential factors influencing fairness in quantized models. The results indicate that model architecture, optimization algorithm, and hardware selection all have potential impacts on fairness (see Tables 1-3 in the link). Specifically, the VGG architecture, Mini-batch SGD, and Ada L4 GPU tend to exacerbate unfairness in quantized models.\n- Q2: We believe there might be a misunderstanding concerning the purpose of the ablation study in Table 3. The primary aim of this ablation is to investigate how the choice of 𝑛 influences the mitigation of unfairness. In our experiments, the strategies with $n=20$ and $n=3$ proved ineffective, resulting in outcomes closely similar to the baseline reported in Table 1. In contrast, the strategy with $n=10$ closely approach the optimal performance of the proposed method (refer to the results labeled 'RE' in Table 2), effectively mitigating unfairness while narrowing the fairness gap between PTQ and QAT. This sensitivity to the choice of $n$ can be attributed to the impact of mask size on the quality and diversity of augmented data. Larger mask sizes, such as $n=20$, may excessively obscure critical image features, impairing the model's ability to learn. Conversely, smaller mask sizes, like $n=3$, might not provide sufficient diversity to capture the inherent variability of the data. Therefore, an intermediate value like $n=10$ strikes an optimal balance, enhancing both the quality and diversity of augmented data, which leads to more robust and effective fairness improvements. To further substantiate our findings, we have conducted a finer-grained sensitivity analysis on mask size $n$ over the range {$3, 5, 8, 10, 12, 15, 20, 30, 40$} (see Table 4 in the link). We have revised the manuscript to provide a detailed discussion of these results, aiming to clarify any potential misunderstandings.\n- Q3: Thank you for your insightful feedback regarding Tables 2 and 3. To address this, we have revised Tables 2 and 3 in our manuscript to include the baseline model results, facilitating clearer and more direct comparisons.\n- Q4: Great thanks for your positive feedback on the ablation study presented in Table 3. We would like to clarify that the ablation study in Table 3 is not designed to identify the optimal $n$, but rather to validate the superiority of a dynamic, randomized approach over static configurations. Specifically, this ablation demonstrates that the random selection strategy for the mask size $n$—defined within the range {$3, 4, \\dots, 20$}, as derived from the optimal configuration in [1]—outperforms fixed choices for $n$.​ Since the selection strategy for appropriate mask size $n$ directly impacts the quality and diversity of augmented data, this findings emphasize that, beyond the amount of augmented data, the quality of augmentation also plays a crucial role in mitigating unfairness. Our results indicate that the random selection strategy not only increases the volume of augmented data but also enhances its diversity, leading to more robust and effective fairness improvements. In contrast, fixed choices for $n$ tend to fail in capturing the inherent variability in the data, resulting in less stable and suboptimal fairness improvements. To further support our conclusions, we have conducted a finer-grained analysis on fixed choices of mask size $𝑛$ over the range {$3, 5, 8, 10, 12, 15, 20, 30, 40$} and compared it to the random selection strategy from the range {$3, 4, \\dots, 20$} (see Table 4 in the link). The results show that the random selection strategy is indeed the most effective in mitigating unfairness. We have revised the manuscript to clarify this point more explicitly and have included the additional experiments in the appendix.\n\nWe truly value your feedback and are deeply grateful for your continued support.\n\n[1] Random Erasing Data Augmentation, AAAI 2020.", "claims_and_evidence": "convincing", "methods_and_evaluation_criteria": "make sense", "theoretical_claims": "correctness", "experimental_designs_or_analyses": "soundness", "relation_to_broader_scientific_literature": "unbalance data, model compression", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The content of this article is very interesting, however I would suggest that the title be more strictly limited, perhaps in the field of face recognition, racial differences, etc. In addition, the means to reflect the article are solved through data enhancement, that is, the title or abstract needs to reflect the limitations of the datasets used.", "other_comments_or_suggestions": "My concern is that \n1. the takeaway2is obvious such as \"Class imbalance is to blame for unfairness\" in line 275 and might not contributes significantly.\n2. Takeaway 3 \"Although quantization-aware training always provides a better overall performance guarantee, deterioration in fairness induced by imbalanced datasets towards protected attributes is much more severe\" is  interesting enough but the problems that were found did not seem to be well addressed. Case in tabel 3 when $n$=20 seems to be similar to baseline in Tabel 1.", "questions_for_authors": "3. Tabel 2-3 lack the baseline model (Tabel 1) for comparison, making it not straightforward\n4. Tabel 3 present a good ablation of problems that were found did not seem to be well addressed, which is not vary stable with different $n$, which should be further discuss.", "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": "", "extra_scores": {}}, {"review_id": "LqhX0N5XxX", "reviewer": "Reviewer_QYq3", "summary": "This work unveils the potential risk of exacerbating the unfairness in model accuracy among various groups. By theoretical analysis and empirical experiments with both Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT), this work identifies several observations, including group White has less performance drop and PTQ behaves better in preserving fairness. The author also verifies several unfairness mitigation schemes, including geometric transformations and random erasing, and demonstrates that these data augmentation techniques could help mitigate the unfairness caused by quantization.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for kindly evaluating that \"the key contributions are novel.\" We also sincerely appreciate your constructive suggestions, and believe that the additional experiments and explanations can address your concerns. The new experimental results are available at https://anonymous.4open.science/api/repo/Rebuttal-30C1/file/Reviewer%20QYq3.pdf?v=f7137c6b.\n- Typos: Thank you for your thorough review and for pointing out these formatting and typographical issues. We have made another careful polish to the paper. \n- Issue 1: To enhance clarity, we specify that our work adopts the quantization method described in [1], which serves as a standard approach in both PyTorch and TensorFlow. This method is widely used in practice and aligns with the default settings in major deep learning frameworks. As suggested, we have also added references to specific quantization methods, including PACT/LSQ for QAT and Q-drop/BRECQ for PTQ, in the Related Work section to improve clarity. Moreover, our theoretical analysis follows general quantization principles and is applicable to different quantization methods (see Table 1 in the link). Specifically, in the final step of the derivation in Theorems 4.1 and 5.1, we only need to replace $\\Vert \\Delta w \\Vert$ with the corresponding quantization error of the specific method—for instance, when applying BRECQ[2], one would replace $\\Vert \\Delta w \\Vert$ with the Frobenius norm of the difference between the full-precision weights and the quantized weights. \n- Issue 2: We would like to clarify that our theoretical analysis for weight-only quantization (WOQ) can be directly generalized to weight-activation quantization (WAQ). This generalization is supported by the findings in [3], specifically Theorem 1, which shows that the impact of activation quantization on the target loss can be transformed into a similar effect as weight quantization. Based on this, we have derived the theoretical bounds for WAQ, which reveal that the upper bound of the excessive loss $G(a)$ introduces two additional terms compared to WOQ: $\\sqrt{n}\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}\\cdot\\Vert g_{w^*}\\Vert+\\frac{1}{2}n\\left(\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}\\right)^2\\cdot \\text{Tr}(H_{w^*})$. The coefficient $\\frac{\\tilde{w}^*_{max}s_{max}}{2+s_{max}}$ has a stronger impact on fairness compared to the WOQ case. To further validate this theoretical analysis, we have conducted experiments with WAQ and the results are consistent with our theoretical findings (see Table 2 in the link). We have included these theoretical analyses, proofs, and the corresponding experimental results and discussions in the revised paper.\n- Q1: Data-free quantization methods with balanced synthetic datasets may help mitigate unfairness caused by data imbalance. As a preliminary exploration, we have conducted experiments with GDFQ[4]. Specifically, for a task involving $n$ classes, the generator uniformly samples labels $y∈${$0, 1, \\dots, n−1$} during training to ensure a balanced class distribution in the synthetic data. The results show that this approach can mitigate unfairness to some extent (see Table 3 in the link). However, it is important to note that data-free quantization is primarily a technique designed to enable quantization in the absence of training data, rather than a direct solution to address data imbalance. In contrast, data augmentation explicitly targets class imbalance.\n- Q2: We acknowledge that the variation in fairness metrics across different models suggests that, in addition to the two key factors identified in our theoretical analysis, fairness in quantized models may also be influenced by other potential factors, including model architecture. To further study this, we conducted additional ablation experiments to explore more factors affecting fairness in quantized models. Our results indicate that model architecture, optimization algorithm, and hardware selection all play a role, with the VGG architecture, Mini-batch SGD, and Ada L4 GPU particularly exacerbating unfairness (see Table 4-6 in the link). In particular, under int 4, VGG exhibits more severe unfairness than ResNet due to its architectural characteristics. Lacking residual connections, VGG is more prone to gradient vanishing and exploding under quantization, amplifying subgroup gaps. What's more, VGG’s uniform layer structure accumulates quantization errors more significantly, whereas ResNet’s residual connections help mitigate these effects by preserving information flow.\n\nYour insights are valuable to us, and we sincerely appreciate your reconsideration of our paper.\n\n[1] Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference, CVPR 2018.\n\n[2] BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction, ICLR 2021.\n\n[3] QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization, ICLR 2022.\n\n[4] Generative Low-Bitwidth Data Free Quantization, ECCV 2020.", "claims_and_evidence": "Most claims made in the submission are well supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense for the problem or application.", "theoretical_claims": "I have checked the correctness of the proofs for theoretical claims in the main text and did not find any issues.", "experimental_designs_or_analyses": "I have checked the soundness/validity of experimental designs. Some issues include:\n\n- The work broadly uses QAT and PTQ to represent the quantization methods used in the analysis. However, these are two major categories instead of the concrete techniques. For QAT, I know from section 3.1 that signed symmetric uniform weight-only quantization, but it's not sure what concrete settings are the author following. I recommend adding the reference to specific methods, such as PACT/LSQ for QAT, or Q-drop/BRECQ for PTQ.\n- In addition, this work only verifies the fairness problems under weight-only quantization settings. The generalization of the conclusion on weight-activation quantization remains unknown.\n\n[1] PACT: Parameterized Clipping Activation for Quantized Neural Networks, arxiv 2018\n\n[2] Learned Step Size Quantization, ICLR 2020\n\n[3] QDrop: Randomly Dropping Quantization for Extremely Low-bit Post-Training Quantization, ICLR 2022\n\n[4] BRECQ: Pushing the Limit of Post-Training Quantization by Block Reconstruction, ICLR 2021", "relation_to_broader_scientific_literature": "This is a brand new direction and research question. The key contributions are novel, while some fairness metrics and mitigation schemes follow previous work and are cited.", "essential_references_not_discussed": "As many representative works in quantization and fairness are cited, I think the related works section in this paper are essential to understanding the key contributions of the paper.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "- The title on each page should be revised to be the title of this paper instead of the placeholder \"Submission and Formatting Instructions for ICML 2025\"\n- Some typos could be fixed (Table 1 under QAT int8 ResNet18 CIFAR-10  \"01.3\" $\\rightarrow$ \"1.3\", line 317 \"is\" $\\rightarrow$ \"are\", line 423 \"adopted on for\" $\\rightarrow$ \"adopted for\", )", "questions_for_authors": "- If the data imbalance is to blame for the unfairness, would data-free quantization methods with balanced synthetic datasets solve the problems?\n- From the results shown in Table 1, fairness metrics variation across different models is more significant than in different quantization settings (especially on VGG19). Any ideas or discussions on this phenomenon?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper studies Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT) to understand how they affect model accuracy across different groups of datasets. It provides both theoretical analysis and empirical verification. The reviewers found the results interesting but provided some comments, especially on the experimental settings and presentation, which need to be addressed.
Decision: Accept (poster)