Empirical Likelihood for Fair Classification
Authors: Pangpang Liu
Yichuan Zhao
Venue: ICLR 2024 poster (2024)
Keywords: Fair Classification
Empirical Likelihood
Primary area: societal considerations including fairness, safety, privacy

Abstract: Machine learning algorithms are commonly being deployed in decision-making systems that have a direct impact on human lives. However, if these algorithms are trained solely to minimize training/test errors, they may inadvertently discriminate against individuals based on their sensitive attributes, such as gender, race or age. Recently, algorithms that ensure the fairness are developed in the machine learning community. Fairness criteria are applied by these algorithms to measure the fairness, but they often use the point estimate to assess the fairness and fail to consider the uncertainty of the sample fairness criterion once the algorithms are deployed. We suggest that assessing the fairness should take the uncertainty into account. In this paper, we use the covariance as a proxy for the fairness and develop the confidence region of the covariance vector using empirical likelihood \citep{Owen1988}. Our confidence region based fairness constraints for classification take uncertainty into consideration during fairness assessment. The proposed confidence region can be used to test the fairness and impose fairness constraint using the significant level as a tool to balance the accuracy and fairness.   Simulation studies show that our method exactly covers the target Type I error rate and effectively balances the trade-off between accuracy and fairness. Finally, we conduct data analysis to demonstrate the effectiveness of our method.

Published as a conference paper at ICLR 2024
EMPIRICAL LIKELIHOOD FOR FAIR CLASSIFICATION
Pangpang Liu
Yichuan Zhao
Mitchell E. Daniels, Jr. School of Business
Department of Mathematics and Statistics
Purdue University
Georgia State University
West Lafayette, IN 47907, USA
Atlanta, GA 30303, USA yi chuan gsu. edu
11u33648purdue.co edu
ABSTRACT
Machine learning algorithms are commonly being deployed in pocision-makingg systems that have a direct impact on human lives. However, if these algorithms are trained solely to minimize training/test errors, they may inadvertently discrim- inate against individuals based on their sensitive attributes, such gender, race or age. Recently, algorithms that ensure the fairness are developed in the machine learning community. Fairness criteria the fairness, but they often use the point estimate to assess the fairness and fail to consider the uncertainty of the sample fairness criterion once the algorithms are deployed. We suggest that assessing the fairness should take the uncertainty into account. In this paper, use the covariance develop the confidence region of the covariance vector using empirical likelihood (Owen, 1988). Our confidence region based fairness constraints for classification take uncertainty into consideration during faimess assessment. The proposed con- fidence region can be used to test the fairness and impose fairness constraint using the significance level as a tool to balance the accuracy and fairness. Simulation studies show that our method exactly covers the target Type I error rate and effec- tively balances the trade-off between accuracy and fairness. Finally, we conduct data analysis to demonstrate the effectiveness of our method.
applied by these algorithms to measure
proxy for the fairness and
INTRODUCTION
Machine learning methods are increasingly used in various domains assist human decision- making. However, concerns have been raised about the fairness of these algorithmic decision- making methods. To address this issue, recent studies have proposed mechanisms ensure the fairness of algorithmic decision systems. One such measure of fairness proposed by Zafar et al. (2017) is the covariance between sensitive attributes and the distance between the subjects' feature vectors and the decision boundary of the classifier. Zafar al. (2019) introduced the covariance measure decision boundary unfairness to design classifiers that are free of disparate impact and disparate mistreatment, while Nanfack et al. (2021) extended this method to decision trees. Other methods to address fairness concerns include Rényi correlation proposed by Baharlouei et al. (2020), covariace operator used by Pérez-Suay al. (2017) measure the dependence between pre- dictions and sensitive variables, and equalized correlations proposed by Woodworth et al. (2017) as fairness constraints. However, these methods often estimate the dependence between sensitive variables and predictions by point estimates, and do not consider the uncertainty of the estimation. Moreover, some measures of fairness are not dimensionless, making it difficult to set the threshold of the fairness constraint.
We have found solutions to these issues through the use of empirical likelihood (EL.) This non- parametric method does not require assumptions about the underlying distribution and has been applied in many problems (Kallus Uchara, 2019; Karampatziakis et al., 2020; Dai et al., 2020; Alemdjrodo & Zhao, 2022; Liu & Zhao, 2023b). In this work, instead of focusing on finding new fairness criterion, we adopt the covariance as a tractable proxy to measure the fairness (Zafar et al., 2017), and study the problem of group faireess, which focuses reducing the difference of favorable outcomes proportions among different sensitive groups The covariance measure can han- dle both discrete and continuous sensitive attributes. Using El., we develop confidence region for

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the fairness criterion. By using EL-based confidence region, we can consider the uncertainty of the estimation of dependence between the sensitive variable and the predictions, which is not possible with methods that rely solely on point estimates. We utilize the significance level (a E [0. 1) as threshold of the fairness constraint, which is dimensionless and provides a clear way to assess the assess the fairness of features by checking whether is
fairness. By the confidence region, included in the confidence region with Type I error rate controlled, and impose the fair constraint by constructing the domain of the parameters of interest such that 0 is included in the confidence region of the covariance vector. We do not need to set the threshold of the covariance vector, and hence the dimension of covariance has no effect on our method. In our algorithm, we balance the trade-off between the accuracy of the estimation of the parameters and the fairness constraints by the significance level. the simulation studies, our method performs good in terms of the coverage probability and the average length of the confidence interval of the covariance. Also, our method performs better than the method in Zafar ct al. (2017) in regard to the trade-off between the accuracy and fairness in some cases. In the real data analysis, we show that our method is robust to distribu- tion shifts and can achieve simultaneous fairness by considering discrete and continuous features as protected attributes. A summary of our contributions is follows.
(1)
We propose an EL-based estimator of covariance, and establish that its limiting distribution fol- lows the standard x² distribution. This conclusion does not require assumptions about the under- lying distribution. Based on this limiting distribution, we can construct a confidence region for the covariance between sensitive attributes and the decision boundary, which can be used in statistical hypothesis test for group faimesss
(2) We formulate the faireess constraint using a confidence region, where the significance level a e [0, 1] serves as the fairness threshold. This approach is more practical than dimensional fairness thresholds proposed in previous literature. The significance level o, which ranges from 0 to mentioned above, can be easily determined in practice. This approach allows the fairness of both categorical and continuous features, and our algorithm can handle multiple sensitive features simultaneously achieve the fairness. By incorporating uncertainty into the fairness measurement, our confidence region makes the fairness criterion more robust and reliable.
measure
(3) Our work introduces a new direction for fairness constraints. We consider uncertainty when as- sessing and imposing fairness, which is often neglected in existing fairness criteria. Our frame- work can be extended to other fairness criteria, such as Rényi correlation (Baharlouei et al., 2020). We provide discussion of applying EL to general fairness measures in Section 6.
The remainder of this paper is organized as follows. In Section 2, we review related work. In Section 3, we review the covariance as fairness constraint, and develop the confidence region for the covariances by EL and present the EL framework for fairness. In Section 4, simulation studies are conducted to evaluate our method and compare with other method. In Section 5, we assess our method by real data analysis. In Section 6, we conclude our work and discuss some potential extensions. The proofs and additional results are included in the appendix.
RELATED WORK
At a high level, our approach develops a confidence region for the fair criteria with the covariance as a proxy of fairness. Our framework enables auditors to assess and enforce fairness in a statistically principled manner, and can be used for statistical inference on group fairness.
Previous literature has studied statistical inference for group fairesss by deriving limiting distribu- tions of relevant statistics. However, these limiting distributions contain unknown values, rendering hypothesis tests impractical. For instance, Besse et al. (2018) showed that an estimator of disparate impact converges to a standard normal distribution that can be used to test for group disparate im- pact. However, the test statistic includes the unknown covariance. Similarly, Si et al. (2021) tested for group fairness using optimal transport projections and studied the asymptotic behavior of the projection distance, but the limiting distribution contains unknown values. Taskesen et al. (2021) developed a limiting distribution based on the concept of Wasserstein projection for testing group faimess, but this distribution also contains unknown values.
is worth noting that some literature has studied inference for individual faimesss Maity et al. (2021) developed the mean of the ratio of two loss functions test statistic for individual faimness.
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However, the test statistic is not well-defined when the classifier correctly classifies. They thus used the ratio of means (instead of the mean of the ratio) as a test statistic, which is not guaranteed to have a limiting distribution. Similarly, Xue et al. (2020) found that the asymptotic distribution for individual fairness depended on an unknown parameter.
Previous hypothesis testing methods have been problematic in practice, as the test statistics lim- iting distributions have not been well-defined. Researchers have thus resorted to plugging estimates into the limiting distributions, making the tests inaccurate. Our approach, on the other hand, em- ploys empirical likelihood ratio, which has a limiting standard x² distribution and does not include any unknown values. As such, can serve as a reliable test statistic for group faimesss.
STATISTICAL FRAMEWORK
In this section, we first provide brief overview of using covariance fairness constraint in previous literature, and highlight its limitations. Then, we introduce our proposed method, which constructs a confidence region for the covariance vector using empirical likelihood, and uses it as fairness constraint.
3.1 REVIEW OF COVARIANCE FAIRNESS CONSTRAINT
The covariance has been used proxy for fairness in previous literature (Zafar al., 2017: 2019; Zink& & Rose, 2020). In this section, we provide a brief overview of fair classification with covariance constraints, and highlight some of the issues that arise when working within this framework.
In binary classification problems, the goal is to find a mapping between feature vectors E R and class labels y 13. For margin-based classifiers, this typically involves constructing decision boundary in the feature space that separates features based on their class labels. Let 0 E O be the parameters of interest, and (z(x) decision boundary. Our objective is to find 0 that results in fair classifier. The main idea is adding a constraint during the model training process. Let & E Ihm be the set of sensitive attributes, such as age or gender, and (()) be the loss function. We denote dg E Ihm as the covariance vector between s and dø(x), and define μta = B(s) and μi = ((dg(z) (x)). By the definition of covariance, we have
be the distance from the feature vectors the classifier
0g = E (s (de - [e]]--
Note that t contains m elements, each of which corresponds the covariance between each sensitive feature and the classifier decision boundary. Let {æ,, 84, W be independent and identically distributed samples. The sample version or is
oo >
-
2 sdd (x:) de), with s
Σ
n
s; and dg
a Σ do (x.)).
n
n
Zafar et al. (2017) estimated the parameters 0 of the decision boundary by minimizing the loss function over the training set, subject to fairness constraints. Specifically, they solved the following optimization problem,
min L(O) BE0
n (d c,
Σ 8: n1
1
( -,
where e RT is the covariance threshold. The fairness constraint in (2) uses the covariance thresh- old E I to limit the covariance between sensitive attributes and the decision boundary, thereby balancing fairness and accuracy.
However, this approach has several limitations. Firstly, the value of 1/n Ei, 3) (æ;) de- pends on the samples, while fixed. Consequently, this method fails to account for the uncer- tainty in the covariance estimation in the faimess constraint. Secondly, since the covariance is not scale-free and с is a m dimensional vector, determining the threshold e that balances accuracy and fairness becomes difficult. Thirdly, when system of m equations and d parameters, which can possibly only have solutions = 0 if m 2 d. Lastly, the confidence region of the covariance vector is not constructed, and statistical inference not addressed this existing framework. To address these limitations, we propose a confidence region-based fairness constraint for classification using empirical likelihood.
0 and dø(x) 0 æ. the constraint reduces to
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3.2 EMPIRICAL LIKELIHOOD FOR COVARIANCE
In this section, we develop the confidence region for the covariance vector de by empirical likeli- hood. Empirical likelihood is used construct generalized likelihood ratio test statistics and cor- responding confidence regions without specifying parametric models for the data. Comprehensive reviews about empirical likelihood can be found Owen (2001) and Liu & Zhao (2023a).
We apply the EL method based on influence functions, which avoids the involvement of nuisance parameters and achieve a limiting distribution without any unknown quantities. For fixed 0, by Equation (1), simple algebra yields that
Vnlão 00) vn 1
si (x)() - do} -0e
-el- ((s (=)(de(x.) peo - -oe VTi(B - -ma) ue)
(3)
((0 (a)dd((s)) uo) - [00 + Op()))
In the last equation, the second term is (1) because vn(s us) = -0)=Opp O1()) and vn(dg - μo) (1) by the central limit theorem. Equation (3) implies that the i-th influence function of σe is (s, - (e)) (d,)
ag. Motivated by Zheng et al. (2012), we construct the empirical likelihood ratio ore based on the estimated influence function follows.
Ep= Σ p. (x, - del ee. 2 =0.
R(o sup
II I (mp.))
(4)
Pl
i=1
where pj is the probability placed on the i-th sample. Using the Lagrange multiplier method to solve (4), obtain
p. n A1++AT((8 (x.) - døl- o
where is the Lagrange multiplier. Hence, the empirical log- ligelikloooodd ratio can be obtained as follows,
22 R(oo)= 2 log(1 + A AT[(s (x)() de) -oel).
(5)
where À satisfies
s () (x:) dgl 00 0. 1+ XT ((: 8) (2:)) do)
(6)
-ao)
i=1
Now, we establish the Wilk's theorem for the empirical likelihood ratio follows (Zheng et al., 2012).
Theorem 3.1 Under conditions C1-C4 in Appendix A, for a fixed 0, at the corresponding true value Tgg have
-2g R(oo) D4 x2.
where X is the chi-squared distribution with m degrees of freedom, and D, denotes the conver- gence in distribution.
Theorem 3.1 provides the asymptotic property of the log-empirical likelihood ratio. The limiting distribution of 2 log R( is standard x2 distribution without any unknown quantities. In con- trast, previous fairness literature (Besse al., 2018; Xue al., 2020; et al., 2021) derived the limiting distributions that contain unknown quantities, which can reduce the accuracy of statistical inference when the estimates of the unknown parameters are used.
We derive the confidence region of de based on Theorem 3.1 for fixed A. The EL confidence region with (1 - a: confidence level for σe at given 0 is constructed as
JEL(ON) (ōr ET 2 log R(ō) < X2n (a)).
where x2 (a) is the upper a- a-quantile of 2 Equation (7) can be used to impose a faimness constraint during the model training process and forms the basis for the fairness inference.
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3.3 FAIRNESS VIA EMPIRICAL LIKELIHOOD
In this section, we develop a novel framework to address the limitations of previous methods and achieve the fairness in classification. We formulate the EL-based fairness framework. Our frame- work can be used for statistical inference and as well for imposing faimess constraint during the model training process.
To conduct statistical inference for fairness, we utilize the confidence region defined in Equation (7). We test for faimess by checking whether included in TEL (oo) for statistical inference. If o E (oe) i.e., -2 log R(O) < (a), it means that O is in the (1 a) confidence region of au and correspondingly, the linear dependence between the sensitive features and the decision boundary is not significant at the significance level.
To obtain a fair classifier, we can impose the fairness constraint during the training process by find- ing the decision boundary parameter 0 such that 0 is included in (gz( Our approach differs from previous literature as we reduce the uncertainty associated with the point estimate by incor- porating the confidence region into the training process. To find the optimal decision boundary parameters that satisfy the fairess constraint, we minimize the corresponding loss function L( over the training under the fairness constraints given by the following optimization problem,
min I(0) s.t. 2 @EO.XER"
log + A (s; ()))(( dol} S Xm
1=1
(8)
(s (o) - de) 0. X (8: do (z;) dg
The constraint in (8) is obtained by plugging = 0 into the equations (5) and (6). Intuitively, as the value of a decreases, the corresponding X (a) increases, resulting a looser fairness constraint and higher accuracy of parameter estimation. Conversely, a larger value leads to stricter fairness constraint, but lower accuracy of parameter estimation. In the extreme case a = 0, T TEL (r) becomes the entire space IR" which means there is no fairness constraint. Thus, the significance level can serve as a trade-off indicator between accuracy and fairness, and unconstrained model can be viewed as special case of our method. Note that the constraint in (2) is symmetric, while the constraint in our framework asymmetric, which contains more information. Our framework uses the covariance faireess proxy. and it can be easily extended to other fair criteria by developing a confidence region for those criteria using EL. We present the use of our approach with the binary logistic regression in Appendix B as example.
SIMULATION
In this section, we evaluate our method by simulation studies. We first examine the limiting distri- bution provided in Theorem 3.1 empirically by the coverage probability and the confidence interval. Then, we explore the trade-off between the accuracy and fairness using our method and compare with the method in Zafar al. (2017) by experiments.
4.1 COVERAGE PROBABILITY AND CONFIDENCE INTERVAL
In this section, examine the limiting distribution from Theorem 3.1 terms of the coverage prob- ability and the confidence interval. We consider a linear decision boundary dg(x)) = 05æ1 + 02 with 0 (01, f2)TT One sensitive feature N NO, 2), while and I2 are both from the Gaussian distribution N N0, 1). The covariance between and 21 is 1/2, the covariance between and I2 is 1, and the covariance between 31 and 22 is O. The sample size n varies from 100 to 1200, and we evaluate the confidence interval for the covari- ance dg between and (2) using EL at a = 0.05, under four scenarios: = (1/ /4 1/3) with oa 11/24. 0 (1/2, 2/3) with dg = 11/12, 0 (2, 1) with de = 2, and 0 = (3, 2) with Te = 7/2. We repeat each experiment 2000 times, and present the coverage probability (CP), the average lower bound (LB). the average upper bound (UB) and the average length (AL) of confidence
is continuous and follows Gaussian distribution
interval for og in Table 1. The results for the scenarios, 0 = (2, 1)" with og = 2, and 0 = (3, 2)
with d = 7/2 are shown in Appendix C.
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Table I: Coverage probability and confidence interval
0
(1/4,
1/3)
(1/2,
2/3)
CP
LB
UB
AL
CP
LB
UB
AL
100
0.935
0.330
0.621
0.291
0.935
0.661
1.243
0.582
200
0.941
0.363
0.570
0.207
0.941
0.726
1.139
0.413
500
0.943
0.396
0.527
0.131
0.943
0.791
1.053
0.262
800
0.947
0.408
0.512
0.104
0,947
0.816
1.023
0.207
1200
0.944
0.417
0.501
0.085
0.944
0.834
1.003
0.169
The results in Table 1 indicate that the coverage probabilities under different scenarios are close to 0.95. Under different scenarios, the true values of σe are between the lower bound and the upper bound at different sample size n, and the average length for dg decreases the sample size n increases. In summary, our EL-based method provides reliable confidence intervals for the covariance.
4.2 TRADE-OFF BETWEEN ACCURACY AND FAIRNESS
In this section, we investigate the trade-off between accuracy and fairness, and compare our ap- proach with the method in Zafar et al. (2017). To facilitate the comparison, we employ the settings proposed by Zafar et al. (2017). We generate 2000 binary class labels uniformly random and assign a 2-dimensional feature vector to each label by drawing samples from two distinct Gaussian distributions: p( y 1) N [[2; 2], [5, 1; 1; 51) and p(zly -1) (N[[2; -2], [10, 1; 1, 3)). We use æ' = [cos(e), sin sin( ()), cos( (b) as a rotation of the feature vector and draw the one-dimensional sensitive attribute from a Bernoulli distribution, p(s = 1) p(
1)/[p(x'|y = 1) + p(x'1) ly -)]. The value of & controls the correlation between the sensitive attribute and the class labels. We choose e = 4/3, a 0.05.
Table 2: Trade-off between model performance and faimess
Zafar et al. (2017)
EL-based fairness
e/a
ACC
F1
p%
DP
FO
ACC
p%
D)
FO
LB
UB
0
0.825
0.827
92.308
-0.029
0.083
0.847
0.845 68.959
0.188
0.366
0.199
0.474
0.1
0.842 0.840 78.548
0.094
0.240
0.843
0.843 81.346
0.065
0.207
-0.069
0.188
0.2
0.840 0.838 73.734
0.124
0.299
0.842
0.842 83.077
0.050
0.188
-0.084
0.171
0.3
0.843 0.841 68.821
0.190
0.366
0.840
0.841 83.654
0.050
0.182
-0.093
0.159
0.4
0.847 0.845 68.959
0.188
0.366
0.833
0.834 87.016
0.025
0.142
-0.101
0.150
0.5 0,847 0,845 68.959
0.188
0.366
0,835
0,836 87.604
0,011
0.135
-0.107
0.142
0.6 0.847 0.845 68.959
0.188
0.366
0.833 0.835 89.368
-0.004
0.116 -0.113 0.135
0.7 0.847 0.845 68.959 0.188
0.366
0.832
0.833 89.941
-0.016
0.109 -0.118 0.129
0.8 0.847 0.845 68.959
0.188
0.366
0.830
0.832 90.533
-0.016
0.103
-0.123 0.123
0.9 0.847 0.845 68.959 0.188
0.366
0.827 0.829 91.716
-0.016
0.090 -0.127 0.117
We partition the data into a training set (70%) and test set (30%) and fit logistic model (Appendix B). We use accuracy (ACC), Fl-score (FI) as the performance metrics the model, p%,-ule' (p%), demographic disparity (DP) and equal opportunity (EO) the fairness metries. Table 2 displays the relationship between the model performance and fairness at various fairness thresholds using the method in Zafar al. (2017) and method EL-based fairness. The first column in Table 2 is the parameter e for Zafar et al. (2017) and the parameter for our method. The results are calculated from the test data, and for method, also calculate the lower bound (LB) and the upper bound (UB) of the covariance. We draw the following conclusions from Table 2:
(1) For the method in Zafar et al. (2017), the threshold e is defined in (2). As e increases, the model becomes more accurate but less fair. As surpasses 0.4, accuracy and faimess remain stable,
Accordiig to the p%orule (Biddle, 2005), the ratio between the proportion of subjects with certain sen- sitive attribute value assigned the positive decision outcome and the proportion of subjects without that value who receive the positive outcome should not he less than p/ 100. The 80%-rule is generally used in practice

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Figure 1: Training on one state (ID (state) evaluation) and testing on other 49 states (OOD evalua- tion) for the income task. The z-axis represents the accuracy. Top lines: The y-axis represents the demographic disparity. Bottom 2 lines: The y-axis represents the equal opportunity.
indicating that 0.4 is an upper bound for e as a trade-off indicator in this scenario. As c can take any non-negative value, finding a suitable threshold in practice is challenging.
(2) For our approach, we use the significance level balance the accuracy and faimess. The value of a varies from 0 to 1, which is easy to control. When O, the model is unconstrained, and 0 is not between the lower bound and the upper bound. As a increases, the model becomes more fair but generally less accurate. The results show that balances the accuracy and fairness effectively. Our method provides the confidence interval, which can used in statistical inference.
Because of the existence of the trade-off between the performance and fairness, model is deemed superior it achieves a better performance and higher fairness than others. Our method achieves better model performance and higher fairness than the method in Zafar et al. (2017) in the follow- ing cases: = 0.1 vs. = 0.3, a = 0.2 vs. = 0.1. and a = 0.3 vs. c = 0.2, a = 0.1 vs. c 0.1, = 0.1 vs. c 0.2, and a 0.2 vs. C 0.2. Table reveals no cases in which the method in Zafar et al. (2017) performs better.
DATA ANALYSIS
In this section, we conduct experiments to assess the efficacy of our method using real data. Firstly, we apply our method on the ACS PUMS datasets (Ding et al., 2021), which encompass distribution shifts, and find our method is robust distribution shifts. Secondly, we consider multiple sensi- tive attributes of the German credit dataset (Dua & Graff, 2019) and show that our method achieves simultaneous fairness.
5.1 ROBUST TO DISTRIBUTION SHIFTS
We leverage the ACS PUMS datasets, which explicitly encompass distribution shifts, where the distribution of the test dataset differs from the training dataset. We study the performance of our method under the geographic distribution shift, where we train model on state and test it on another as Ding et al. (2021) did. This dataset is accessible through the "folktables" Python package. In experiments, task: predicting income above 50000. The dataset encompasses 10 features along with a single label denoting income above 50000. The size of the training dataset varies from 471 3 195665.
construct logistic model (Appendix B) for the income prediction
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We designate race the sensitive attribute. In Table 3 and Figure I, present the accuracy (ACC), demographic disparity (DP), and equal opportunity (EO) metrics with respect to race, for the income prediction. Each figure includes two crucial facets: the in-distribution (ID) results, signifying the training outcomes, and the out-of-distribution (OOD) results, indicating the performance across 49 follows.
other states during testing. The insights derived from these experiments
(I) Figure 3 in Ding et al. (2021) underscores marked separation between the in-distribution and out-of-distribution results. This divergence signifies a severe violation in model accuracy and demographic parity. In contrast, results in Figure 1, depict OOD outcomes distributed around the ID results in a random pattern under the demographic parity and equal opportunity metrics. This pronounced alignment strongly signifies the robustness of our method distribution shifts.
(2) Table 3 presents the evaluations for 0.05 and 0.5 using our method. The ID outcomes denote training results, while the OOD represents the averaged evaluations across the 49 states during testing. A comparison between the IDD and OOD results reveals small fluctuations, thereby undertining the robustness of our approach under distribution shifts. Intriguingly, the variation be- tween 0.05 and a 0.5 elucidates a consistent trend: higher fairness at the cost of marginally reduced accuracy. This consistent pattern reaffirms the efficacy of the underlying balancing role of a. Table shows the results without fairness constraint. Upon comparing the results presented in Table 3 and Table 4, we find our method leads to a decrease in both DP and EO in most cases at a minor cost to accuracy.
Table 3: Results using our method
0.05
0.5
ID (training)
OOD (test)
ID (training)
OOD (lest)
State
ACC
DP
EO
ACC
DP
EO
ACC
DP
EO
ACC
DP
EO
CA
0.7254
0.2008 0.2309 0.7287 0.2073 0.2120 0.7243 0.1991 0.2279
0.7284 0.2034 0.2123
TX
0.7348
0.2195 0.2637 0.7346 0. 1.892 0.2085 0.7335
0.2195
0.2621
0.7340 0.1905 0. 091888
DE
0.7324
0.2515 0.2865 0.7396 0.2895 0.2930 0.7286
0.2176 0.2780 0.7383 0.2665 0.2619
NV
0.7209 0.2193 0.2230 0.7334 0.1781 0.1796 0.7167 0.1944 0.2076 0.7319 0.1657 0.1542
KY
0.7468 0.1591 0.1802 0.7225 0.1046 0.1405 0.7456 0.1444 0.1523 0.7194 0.0831 0.1287
FL
0.7403 0.2157 0.2517 0.7315 0.1655 0.1789 0.7383 0.2115 0.2454 0.7301 0.1587 0.1743
MO
0.7466
0.1479 0.2178 0.7260 0.1242 0.1532 0.7460
0.1435
0.2063 0.7233 0.1024 0.1407
NE
0.7346 0.0224 0.1153 0.7168 0.0713 0. 0.1206 0.7311
-0.0137
0.0417 0.7128
0.0436 0.1096
5.2 MULTIPLE SENSITIVE ATTRIBUTES
In this section, we evaluate the performance of our proposed method on a real-world dataset, con- sidering two sensitive features simultaneously. Specifically, we use the German credit dataset (Dua & Graff, 2019), which contains 1000 instances of bank account holders and is commonly used for risk assessment prediction. Each sample is described by 13 categorical, numerical. and binary attribute. Two sensitive attributes, gender and age, are considered in the evaluation. While previous studies have binarized the age feature by thresholding 25 in Kamiran & Calders (2009),
Table 4: Results without constraint
ID (training)
OOD (test)
State
ACC
DP
E0
ACC
DP
EO
CA
0.7559
0.1996
0.2641
0.7296
0.3867
0.3156
TX
0.7538
0.2529
0.3355
0.7424
0.3374
0.3749
DE
0.7382
0.2760
0.3334
0.7419
0.3410
0.3540
NV
0.7477
0.2798
0.3011
0.7400
0.3547
0,4019
KY
0.7540
0.2375
0.3306
0.7388
0.2732
0.3408
FL
0.7565
0.2982
0.3698
0.7419
0.2944
0.3287
MO
0.7511
0.1915
0.2664
0.7375
0.2582
0.3321
NE
0.7439
0.2386
0.4966
0.7345
0.2542
0.3507
8

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Published as a conference paper at ICLR 2024
0.35
0.55
0.0
0.10
Orrdie
0.5
0.85
60.00 01
e0.
0.25
-0.05
-0.08
(a) Covariance contour under unconstrained model
(b) Covariance contour under constrained model
Figure 2: The 95% confidence region for constrained model and unconstrained model. The xraxis is the covariance between age and the decision boundary. The y-axis is the covariance between gender and the decision boundary. The red line represent covariance contour at 95% confidence level.
avoid this step to preserve the information contained in the continuous age feature. To avoid the effect of sample size on the confidence region, we split the dataset into equal training and test sets.
We train logistic model (Appendix B) on the dataset, using both gender (binary) and age (con- tinuous) protected attributes. Without considering faimesss constraints, the model achieves an accuracy of 0.742. When we apply our EL-based fairness constraint, the accuracy decreases slightly to 0.732. Figure 2 shows the 95% confidence region under the constrained and unconstrained mod- els, respectively. We observe that, without the fairness constraint, the covariance vector (0, 0) falls outside the 95% confidence region, while with the faimess constraint, the covariance vector (0, 0) falls within the 95% confidence region. This suggests that our method effectively achieves a higher level of fairness while maintaining reasonable level of accuracy.
DISCUSSION
In this paper, we propose confidence region for the covariance vector between the sensitive features and classifier decision boundary by empirical likelihood. Our method can be used in statistical inference for the group fairness, and imposing a fairness constraint in the process of training model. We can also develop the confidence region of the covariance vector by jackknife empirical likelihood (Jing et al., 2009). Recall that ōg 1/n Σ ( s)(do(z,) (x,))dd)) do). We define the jackknife pseudo
values o, năø (n 1)0,-1) of a for 1,... where o (-)) is calculated by removing the j-th element. Then the jackknife EL ratio at og is defined as
(oo0)) max [y (np))
Σ PjO 00. Σ Pj =
where Py is the probability placed on the j-the element. Notably, the logarithm of the jackknife EL ratio, denoted as -22 log Rt' (ao), may be proved converge to the x3 distribution. In this paper, we use the covariance as a fairness criterion. However, covariance is not a perfect fairness criterion. We can apply EL other non-linear fairness criteria, such as the coss-covariance operator (Pérez-Suay et al., 2017), equalized correlations (Woodworth et al., 2017). the Rényi correlation (Baharlouei al., 2020) and the Hirschfeld Hiischfeld-GGebeeiin-REe Gebelein-Rémn maximum correlation coefficient (Mary et al., 2019). Some valuable references of adapting our framework to non-linear measures are Qin & Lawless (1994); Kremer et al. (2022); Nabi et al. (2022). If the empirical estimating equation contains non- i.i.d. elements, we can develop the confidence region of the fairness measures by using jackknife empirical likelihood (Jing et al., 2009; Zhao et al., 2015; ; Li et al., 2016; Sang et al., 2020; Matsushita & Otsu, 2020; Liu & Zhao, 2023a). We also tackle fairness concerns by incorporating the prior information using Bayesian empirical likelihood (Lazar, 2003; Chaudhuri et al., 2017; Zhao et al., 2020) and Bayesian jackknife empirical likelihood (Cheng & Zhao, 2019).

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Published as a conference paper at ICLR 2024
ACKNOWLEDGMENTS
Yichuan Zhao acknowledges the support from Simons Foundation (Grant Number 638679), and the National Science Foundation (Grant Number 2317533).
REFERENCES
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Bo Dai, Ofir Nachum, Yinlam Chow, Lihong Li, Csaba Szepesvari, and Dale Schuurmans. Coindice: Off-policy confidence interval estimation. In Advances in Neural Information Processing Systems, volume 33, pp. 1398-94111 2020.
Frances Ding, Moritz Hardt, John Miller, and Ludwig Schmidt. Retiring adult: New datasets for fair machine learning. In Advances in Neural Information Processing Systems, volume 34, pp. 1478-6490 Curran Associates, Inc., 2021.
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and 2m + 1 functions as Published as a conference paper at ICLR 2024
A PROOF OF THEOREM 3.1
For case of expressing the conditions needed to derive the limiting distribution of the empirical log-likelihood ratio, we denote the true parameters
ve (o 14 "oo) 2m++
the correspond generic values as
1 (o Mi 12) E 2m+1
(8d)((r) - 111122
8 p1
(x(x)
P2
To establish theoretical results, we assume that the following conditions hold.
C1. EHo(s, a; v) o for v # vo.
C2. Ho(s, x; v) has continuous first-order derivative with respect to
C3. There exist neighborhood V of vo and an integrable function (8, with E[M((s, x)] oc such that
HHo (H, Ov'
sup Ho 2; 0; ((s æ),, sup wEV
S M(s
wEV
C4. 1EHH(( v0 is non-degenerate and (oH a; vo) Ho oz; vo) is positive definite.
The condition Cl ensures vo is identifiable. The conditions C1-C4 guarantee Ho æ;; vo) is well defined. We define (o()) = (a; (x;) dol The proof of Theorem 3.1 is similar the argument of Zheng et al. (2012). To prove Theorem 3.1, we need the following three lemmas from Zheng al. (2012).
Lemma A.1 (Lemma A.I. Zheng et al. (2012)) Under the conditions C1-C4, we have that max ISi5n
gi(ore = op(vn).
Lemma A.2 (Lemma A.2. Zheng et al. (2012)) Under the conditions Cl-C4, have that n -1 -EE g(oo)g(( (e)) converges in probability to E as n + 0o, where is the variance- covariance matrix of g())).
Lemma A.3 (Lemma A.3. Zheng et al. (2012)) Under the conditions CI-C4, we have that 1/2 E g( converges in distribution to (00 Σ) as n + 30.
By the Lagrange multiplier method, we have
Pr 1 + A g((o))
where À satisfies m equations given by
1 n
gi(Or)
((A)
1 gi( 0.
n1
Let = pw where p > 0 and |w]]
1. Let (0)))) = Iax By Lemma A.I, one
has
g0(0)) = (2
(A.1)
We have
0 [I(pw))]
w h(pw))]
(A.2)
z pw Σn w
w Σ (0)))
+ pg" (ge)
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Published as a conference paper at ICLR 2024
where
nn Σ 91 ( n ==1
By Lemma A.3, we have 1/nw T Σi=1 (:(ee) Amin is the smallest eigenvalue of Σ. Combining (A.1) and (A.2). it follows that
T w In w 2 Amin + op)(1, where
|||| Op(n O/(2.
(A.3)
Let X gi( We have
max
14 = Op 1/000 oz)())
(A.4)
Combining (6), (A.3) and (A.4), we obtain,
A
En
(:(00)
+ +z1(1).
(A.5)
71
Therefore,
1
X (o)
9.
(
Σ
)1(00) tp((1).
(A.6)
Vn
i=1
Taking Taylor expansion at x 0, we get from Equations (A.5) and (A.6)
-2 R(oo) 2 Σ
log 1 + A ATAgg gi(0o)}
==1
= 2 >
T 9. T> op(1
(A.7)
i=1
n
a
Di
9i(oo)
Σ
Σ 9( + 0p)
Vn
By condition C3 and Lemma 1, the last term in (A.7) is 0p(1). Thus, we have
2 log R(oo)
- xm
by using Lemmas A.2 and A.3.
B EXAMPLE
As an example, we present the use of our approach with the binary logistic regression.
Example B.1 Binary Logistic Regression. In binary logistic regression, the objective is predict the binary label by the features x. The classifier decision boundary is defined as do(x) = x. The sensitive vector is denoted 8, and the goal is ensure fairness with respect this sensitive attribute vector. The logistic regression model assumes probability distribution as follows,
1
0
p = 1/ 0) = 1 1 1+e-0Ta' p(y = 11x 1
The loss function is defined as -(0)= = - En log ply; x,, 0). Therefore, the fair logistic regres- sion by EL can be obtained by solving the following problem.
min AER,ERR"
log psp((aaa 0)
1=1
s.
log (1 + (8, - 50 x; - [++] < x (a),
1
(8: -0) x; 0 T
+ X (8; ( 0; x. 0.
77
14

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Published as a conference paper at ICLR 2024
C COVERAGE PROBABILITY AND CONFIDENCE INTERVAL
Table 5 shows the coverage probability and confidence interval for the scenarios, 0 = (2, 1)T with σe and 0 (3,2) with de = 7/2.
Table 5: Coverage probability and confidence interval
0 = (2, 1)
(3, 2)
CP
I.B
UB
AL
CP
LB
UB
AL
100
0.933
1.352
2.813
1.461
0.936
2.434
4.846
2.413
200
0.943
1.520
2.557
1.037
0.946
2.708
4.421
1.713
500
0.939
1.684
2.341
0.657
0.939
2.978
4.064
1.085
800
0.943
1.747
2.266
0.520
0.945
3,082
3,941
0.859
1200
0.945
1.791
2.215
0.424
0.943
3.155
3.856
0.701
15


Reviews:
[{"review_id": "02GSwbDB0S", "reviewer": "Reviewer_YzKS", "summary": "This paper proposes a framework to ensure fairness of classifiers while handling the uncertainty from point estimates using finite samples. In particular, formulating fairness in terms of the covariance between sensitive attributes and the decision boundary (Zafar et al. 2017), the authors propose using the empirical likelihood method to derive a confidence region for the covariance vector. Empirical evaluation using simulation as well as real-world data shows that the proposed approach can balance the accuracy-fairness tradeoff and be robust to distribution shifts.", "questions": "1. How does the proposed method compare against Zafar et al. (2017) under distribution shifts?\n\n2. Is the discussion about EL for general fairness measures new or known results? If it is the former, there needs to be more details and proofs. \n\n3. I was not sure about the significance of Figure 1. Also, should the legends ID and OOD evaluations be switched in Figures 1 and 2? Figure 2 was also extremely hard to read.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The consideration of uncertainty in fairness assessment is interesting, and addresses a problem that is underexplored in existing approaches.\n\nThe application of an EL-based estimator for fairness using confidence intervals is a novel contribution. The authors suggest that this approach could be applied to other fairness criteria, which could increase the impact of this method broadly.\n\nThe paper is overall well-written and technically sound as far as I can tell.", "weaknesses": "While the comparison to Zafar et al. (2017) is very thorough, no other fair learning methods are considered in the empirical evaluation. In particular, comparing group fairness metrics such as DP and EO against baselines that directly optimize those notions could serve to more strongly justify the proposed approach using covariance-based fairness constraints. \n\nI was a bit confused about the discussion at the end about applying EL to other fairness criteria. If indeed extending to non-linear measures is straightforward, it is unclear why the current method only considers covariance as a fairness proxy.\n\nThe motivation for the covariance-based fairness constraints was not very compelling. The authors suggest that determining the threshold for accuracy-fairness tradeoff becomes easier with the proposed approach. However, it is still unclear what the appropriate confidence region is for a given group fairness notion (whether a soft constraint to bound the violation or as a hard constraint).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qTVA2ZCKod", "reviewer": "Reviewer_Rn9g", "summary": "Based on the covariance fairness measure by Zafar et al. (2017), the authors propose a fairness constraint based on a statistical likelihood test. Specifically, they formulate a constraint that requires that a statistical test with level alpha does not reject $H_0$: the classifier is fair.", "questions": "Some minor questions and comments:\n- On the bottom of page 3, you state that a system of $m$ constraints and $d$ parameters with $m\\geq d$ can only have the solution $\\theta=0$. If I understand correctly $m$ is the number of sensitive attributes and $d$ is the number of features. Does it make sense that $m\\geq d$?\n- In equation (8), is $\\lambda$ a decision variable of the minimization problem? \n- Am I correct in my analysis that (8) is nonconvex?\n- I think Figure 1 is very hard to understand. Should we compare Figure 1 with Figure 2 and observe that the ID results are more representative of the OOD resutls than in Figure 2?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I think that the idea of using a likelihood statistic for enforcing fairness is novel and an interesting approach. The paper is well written and supported by the required theoretical developments.", "weaknesses": "I think the main limitation of this work is the experimental evaluation. I understand that the method is based on Zafar's covariance fairness measure, and comparing to that is very valuable. However, since this seminal work in 2017, there has been a lot of development in the community regarding fairness regularizers or constraints. Comparing the results of your approach to some of these more recent works would be of interest.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EtJMwudW1d", "reviewer": "Reviewer_umPv", "summary": "This paper proposes to use the covariance as a proxy for the fairness and develop the confidence region of the covariance vector using empirical likelihood. In this way, a confidence region (with the significant level alpha being used for the fairness constraint) can be used to provide a more interpretable way to trade-off between accuracy and fairness (DP).", "questions": "1. Can you provide more simulation results in Table 2 for Zafar et al. with 0<c<0.1? E.g. c=0.01, 0.02, ... Meanwhile, can alpha take values of 1? Or what will happen when alpha is close to 1, e.g. alpha = 0.99?\n\n2. In Table 3, increasing alpha from 0.05 to 0.5 does not seem to affect the performance much (both in training and testing). Is it still the case when alpha is increased to e.g. 0.9?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The proposed approach is technically sound and does seem to have the potential to replace the covariance based measure in Zafar et al., 2017 and become a more general measurement for DP.\n\n2. Comparing results from Table 3 and 4, the proposed algorithm does have a positive impact on both DP and EO on the ACS PUMS datasets including the OOD cases.", "weaknesses": "The proposed method itself is technically sound and has good potential. Most of my concerns are regarding the evaluation of the method in the experiments.\n\n1. The simulation results in Table 2 do not seem to be a fair comparison. I would suggest adding more granularity for Zafar et al. with 0<c<0.1.\n\n2. In Table 3, increasing alpha from 0.05 to 0.5 does not seem to affect the performance much (both in training and testing). \n\n3. The proposed algorithm should be tested on more commonly used fairness datasets such as [1] or the ones from UCI Machine Learning Repository (https://archive.ics.uci.edu/)--- Adult Census Income, German Credit, etc.\n\n4. Only one baseline is compared in the experiments. I would suggest the authors to test against other baseline approaches optimizing for DP.\n\n[1] Ding, Frances, Moritz Hardt, John Miller, and Ludwig Schmidt. \"Retiring adult: New datasets for fair machine learning.\" Advances in neural information processing systems 34 (2021): 6478-6490.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)