Multi-Output Distributional Fairness  via Post-Processing
Authors: Gang Li
Qihang Lin
Ayush Ghosh
Tianbao Yang
Venue: Accepted by TMLR ()

Abstract: The post-processing approaches are becoming prominent techniques to enhance machine learning models' fairness because of their intuitiveness, low computational cost, and excellent scalability.  However, most existing post-processing methods are designed for task-specific fairness measures and are limited to single-output models. In this paper, we introduce a post-processing method for multi-output models, such as the ones used for multi-task/multi-class classification and representation learning, to enhance a model's distributional parity, a task-agnostic fairness measure. Existing methods for achieving distributional parity rely on the (inverse) cumulative density function of a model’s output, restricting their applicability to single-output models.  Extending previous works, we propose to employ optimal transport mappings to move a model's outputs across different groups towards their empirical Wasserstein barycenter. An approximation technique is applied to reduce the complexity of computing the exact barycenter and a kernel regression method is proposed to extend this process to out-of-sample data. Our empirical studies evaluate the proposed approach against various baselines on multi-task/multi-class classification and representation learning tasks, demonstrating the effectiveness of the proposed approach.

Published in Transactions on Machine Learning Research (03/2025)
Multi-Output Distributional Fairness via Post-Processing
Gang Li Texas ABM University
gang-li@lama.ccm eduu
Qihang Lin The University of lowa
seraanggliinnwowweee edu
Ayush Ghosh The University of lowa
apuskghhoshh700@mmall.com
Tianbao Yang
Haanbao-yanggaae edu
Teaas ABM University
Reviewed on OpenReview: https: //
// openreview. net/ forum? id=MJOKrHqiVi
Abstract
The post-processing approaches are becoming prominent techniques to enhance machine learning models' fairness because of their intuitiveness, low computational cost, and excellent scalability. However, most existing post-processing methods are designed for task-specific fairness measures and are limited to single-output models. In this paper, we introduce post-processing method for multi-output models, such as the ones used for multi-task/min mmulti- class classification and representation learning, to enhance model's distributional parity, a task-agnostic fairness measure. Existing methods for achieving distributional parity rely on the (inverse) cunulative density function of a model's output, restricting their applicability to single-output models. Extending previous works, we propose to employ optimal transport mappings to move a model's outputs across different groups towards their empirical Wasserstein barycenter. An approximation technique is applied to reduce the complexity of computing the exact barycenter and kernel regression method is proposed to extend this process to out-of-sample data. Our empirical studies evaluate the proposed approach against various baselines on multi-task//mulle mulltiilassss classification and representation learning tasks, demonstrating the effectiveness of the proposed approach
1 Introduction
In machine learning, multi-oulput learning is broadly defined domain (Xu et al. 2019 Liu el. al. 2018); where the goal is to simultaneously predict multiple outputs given an input, such muulti-lbbe classification, multi-class classification, multi-target regression, etc. In contrast to conventional singlo-output learning like binary classification, multi-output learning is characterized by its multi-variate nature, whose outputs exhibit rich information ion for further handling. Multi-output learning is important for real-world decision-making where final decisions are made by considering and weighting multiple factors and criteria. For example, when applied to college admission, the predicted multi-outputs can represent a prospective student's likelihoods of accepting the offer, needing financial aid, completing the degree, finding job graduation, etc. Those outputs are weighted to guide admission decisions though the weights may vary with colleges, majors and years (Jaschik 2023).
However, multi-output learning for decision-making faces the challenge of bias and fairness. There is plenty of evidence indicating the discriminatory impact of ML-based decision-making on individuals and
groups O'neil] 2017 Datta et al. 2014 Bolukbasi et al. 2016 Barocas Selbst, 2016 Raji Buolamwini
"Cdee available

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Figure 1: Suppose the original model's outputs (left) minimize prediction error and have a good performance, one can balance performance and fairness by applying our method with setting 0.5 (middle) or achieve exact fairness by setting a 0 (right).
2019), such as racial bias in assessing the risk of recidivism (Flores et al. 2016) and gender bias in job advertising (Simonite) 2015). To mitigate the bias in machine learning, numerous fairness criteria and algorithms have been proposed Corbett-Daviees et al.) 2017 Barocas et al. 2023). These methods introduce statistical constraints during training or post-process the predictions to ensure fair treatment in accordance with corresponding fairness notions such as Demographic Parity (Calders ct al. 2009
Chuang & Mrouch,
2021), Equality of Odds or Equal Opportunity (Hardl al. 2016 Awasthi et al. 2020), Strong Demographic
Parity (Agaraal al.) 2019 JJiang
2020) and AUC fairness {Vogel et al.) 2021 Yang al. 20233 Yao
ct al. 2023). Nevertheless, almost all existing methods focus ensuring fairness for binary classification
regression within the context of single-output models. Extending fairness to multi-output settings, such
multi-label/mulliae muuliiclass classification and representation learning, remains underexplored and non-trivial.
A naive approach for multi-output fairness is to apply existing fairness-enhancinn algorithms for single-output models to each output individually. However, removing unfairness in each output may not help reduce the unfairness in the joint distribution of the outputs. Consider a model with two outputs, with the distributions for two groups shown in Figure ! (left). The outputs have the same marginal distribution in both groups, each dimension of the outputs is considered to be fair. However, the outputs have very different joint distributions for different groups, leading to potential unfairness. For example, suppose Outputl and Output2 denote the likelihood of accepting the offer and complcting the degree in college admission procedure, then when a college requires outpuls are higher than 0.45 and 0.6 respectively (shown in dot lines in Figure D it leads to only students in group 1 being admitted. To address the challenge, in this paper, we propose a post-processing method to enhance fairness of multi-output models on multi-group data by producing a similar distribution of multi-dimensional outputs on each group with minimal manipulation(i.e. minimal mean squared error compared with original outputs). Before discussing details, we present 2D example in Figure [
Mcasuring fairness based on the distribution of the multi milti-dimenstional outputs on each group leads to a strong notion called Statistical Parity or Strong Demographic Parity, which has been studied recently for single-output problems literature (Agarwal et al. 2019 Jiang et al. 2020 Chzhen et al. 2020 Hu et al. 2023). We apply the same notion to multi-outpuu problems and call Distributional Parity (Delinition However, extending this kind of fairness from single-output problems to multi-output problems is non-trin non-trivial due to some technical and practical issues: (1) what is the optimal fair predictor for multi-output problems, which achieves Distributional Parity while preserving accuracy best; (2) empirically, how to compute the theoretical optimal fair predictor with only a set of observed data; (3) practically, how to generalize the post-pun prot-processiin method from training data to out-of-sample test data. Our approach addresses the mentioned
challenges and our main contributions are summarized below:
Following Gouic et al. (2020): Chzhen et al. 2020 Chzhen & Schreuder (022)), we derive post-processing method by projecting a multi-output model to a constraint set defined by a fairness inequality that promoles distributional parity among multiple groups up to a user-specified tolerance of unfairness. We generalize the closod-form solution of the projection by Chzhen et al. (2020); Chzhen & Schreuder (2022) from single-output case to multi-output case using the optimal transport mappings to the Wasserstein barycenter of the model's outputs each group.
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The aforementioned closed form involves the density function of the model's outputs, which is unknown in practice. Although it can be approximated using training samples, the computation cost of the Wasserstein barycenter is still prohibitively high. To address this issue, we propose to replace the barycenter in the closed form by a low-cost approximate barycenter from Lindheim (022)) using training data. Additionally, we propose kernel regression approach to extrapolate the closed-form solution, so the post-processing method can be applied to any out-of-sample data.
The proposed method is task-agnostic and model-agnostic. Numerical experiments are conducted to compare our method with various baselines on multi-label / classification and representation learning. The results demonstrate the effectiveness of our method.
2 Related work
The existing methods to achieve algorithmic fairness include three primary categories: 1) pre-processing
methods that exclude sensitive features from data prior to training machine learning models (Dwork et al.,
2012; du Pin Calmon et al.]
2018 Kamiran & Calders
2012 d' d'Alessandro et al. 2017]; 2) in-processing
methods, which attain fairness during the training phase of the learning model (Agarwal al.
2018 Goh
et al.
2016 Zhang et al. 2018 Kim et al.) 2019 Hong & Yang 2021 Du et al. 2021 Mo et al.
202 Yao
ct al.
2023 ;; and 3) post-processing methods designed to alleviate unfairness in the model inferences a after
the learning process (Hardt et al. 2016 Lipton et al. 2018 Cui et al. 2023 Petersen et al.
2021 Lohia
let al., 2019 Xian et al. 2023). The pre-processing methods are inadequate if the sensitive information can
be inferred from the remaining variables. The in-processing methods have high computational cost since they typically require re-training a model when the tolerance of unfairness changes. Our method belongs to the last category, which can be directly applicable to any pre-trained model and thus avoids the high computational cost from re-training a model.
This work is motivated by Gouic et al. 20200 Chzhen et al. (0200: ; Chzhen & Schreuder (2022) where
lask-agnostic post-procsssing methods are developed on the empirical cumulative density function and the quantile function of the output, which is limited to single-output models. We extend their approaches for multi-output models using the optimal transport mappings to the barycenter of the model's outputs on different groups. Different from their methods, we approximate the barycenter to reduce the computational cost and apply kernel regression to extend our post-processing method to any new data.
There also exist post-processing methods for specific machine learning tasks. For example, assuming a Bayes optimal score function is available, the group thresholding methods by Gaucher et al. (2023); Chzhen et al. (2019); Schreuder & Chzhen (2021): Zeng et al. (2022) are developed for single-task binary classification. A similar thresholding mothod has been developed by Denis et al. (2021) for multi-class classification where the thresholds are computed using the Lagrangian multipliers of the demographic parity constraints. The method by Xian et al. (2023) is also for multi-class classification where the output of the score function is mapped class label by the optimal transport mapping to Wasscrstein barycenter with restricted supports. Compared to these works, the method in this paper is task-agnostic and can be applied to multi-task learning and (self-supervisee) representation learning. Hu et al. (023) developed an approach to enforce strong demographic parity in multi-task learning including both regression and binary classification tasks. Their method essentially applies the single-output method by Gouic et al. (2020); Chzhen et al. (2020) to each task separately, which may not guarantee the distributional parity in this paper. See the example in Figure [ Though there large number of litcrature working on learning representations independent of sensitive
attribules, they achieve this by adversarial training (Zhang et al.
2018 Kim et al.) 2019 Xie et al.) 2017
Madras et al. 2018), variational auto-encoders {Louizos et al.) 2015
Sarhan et al. 2020 Amini et al.
2019
Gong et al. 2024h, or contrastive learning et al.) 2023 Qi et al.
2024 Zhang et al. 2022 Park et al.
2022), which are mostly in-processing methods and thus have higher computational cost than our method in general.
The multi-output fairness problem in this paper is closely related to compositional fairness introduced in Dwork & Ilvento| (2018). which highlights that individually fair classifiers may not compose into fair systems. While Dwork & Ilvento (2018) focuses on categorical outputs, examine continuous output distributions.
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Since categorical outputs often result from thresholding softmax on continuous outputs, compositional fairness is special case of multi-output fairness, and our method can directly address it.
3 Preliminaries
We consider a general multi-output machine learning problem, e.g., multi-output classification, multi-output regression and representation learning. Let (X,S) 5) be a random vector, where X E Rd is the feature vector and S E S is a sensitive attribule. It is assumed that S = 'm] := (l....,m). A sample from the distribution of (X is denoted by (x, We denote by f" TRd x S - IR a machine learning model learned to generate a k-dimensional output. For example, f can be multi-task regression model to predict vector of continuous values, multi-class/ aaaaas/sss/m////a/ /wwltmmtaaattt///eee task classification model that returns the probability of each class/task, a feature extractor that outputs a high-dimensionll vector of an input data such as an image.
The focus of this work not to train the model f". Assuming f is already obtained, this paper studies how to measure and improve its fairness using post-processing method to modify the output f (X, S). More specifically, given f", we consider a numerical approach for building a mapping f IR S R that approximates f* but produces an output f() more fair than (x( 5). Here, the approximation error is measured by
R()) E|I (XI," S(X
(1)
If f* performs well for the task it was built for and the performance metric is continuous, small R(f) also ensures / has a reasonably good performance, too.
3.1 Distributional Parity
Many existing definitions and measures of fairness in literature are task including, for example, demographic parity, equalized odds and equal opportunity (Dwork et al.] 2012 Hardt et al. 2016), which are specific to classification problems. In order to be lask-agnostic and model-agnoslic, we are interested in the fairness measure that is applied to the distribution of f(X, S) and fxX S). A measure of this kind is the strong demographic parity introduced by |Agaraal et al. (2019); Jiang et al. (2020); Chzhen et al. (2020), which essentially indicates that a model (x.,s) fair only when f(X, has the same distribution conditioning S for any E S. Although they only consider a single-output model, their fairness measure also applies to the multi-output case. Next we present their fairness measure formally with the name multi-output distributional parity. This new name helps differentiate it from the traditional demographic parity for binary classification (Calders et al. 2009).
Definition 1 (Mulli-oulpul Distribulional Parily) A measurable mapping J Rd x S IR salisfies distribu- tional parity if, for every s, s' e S, f(X, 8) conditioning on S = = and f(X,S) conditioning on S = have the same distribution.
Remark: Although we focus on Desinition I an extension of demographic parity, it is natural to extend other notions of fairness, such as equal odds and equal opportunity, in a similar fashion. In particular, suppose f(X,
S) ) is used to predict a target variable Y which takes values from a finite set J (e.g., multi-class multt-llss/mmultllbell laul classification). We say f satisfics distributionally equal opportunity with respect to a class y E ) if, for
every 8,8' E S, S) conditioning on (S, = (x, and S(X,S) conditioning on (S = (s',y) have the same distribution. Similarly, we say f satisfies distributionally equal odds if, for every s, E S and every y E y, f(X conditioning on (S, Y)= = (s. y) and f(X,S) conditioning on (S,Y) = (s', y) have the same distribution.
Consider a joint distribution of (X,S,Y) where Y E RR a target vector. In the sctting of a single-task binary classification, we have k = 1, p = and Y E [1, is the true class label. Then the model S(X RR typically represents score/probability of positivity, which is used to predict Y by comparing f(X, S) with a threshold 0, namely, the predicted label Ý generated as F =1 if f(X 2 0 and Y=
if S) < 0. The traditional demographic parily (Calders al. 2009 Dwork et al. 2012) holds if P(Ý = 1 = s) = f( = 1|$ = =') Vs, S. Obviously, when satisfics distributional parity, it satisfies demographic parity for any threshold 0. In fact, this is still true for multi-task binary classification problems, where p = k > 1 and Y *
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The notion of demographic parity has been extended for the multi-class classification problem, where k >> 1, = and Y E [k]] (Xian et al. 2023). In this case, each coordinate of f(X,5) represents the score of one class in [c] and the predicted label is typically generated as ÿ arg ma fi(X, S).. According to Xian et al. 2023), model satisfies demographie parity when
PØ = y|S = =) = P( = vls =s) Vs,8' E E y [
(2)
which is clearly implied by distributional parity.
3.2 Wasserstein Distance and Wasserstein Barycenters of Discrete Distributions
It is challenging and, sometimes, unnocessary to obtain model f that exactly satisfies the distributional parity in Definition [ In practice, model slightly violating Definition can be acceptable for some applications. To quantify the extent to which a model f satisfies the distributional parity, a statistical distance is often introduced to mcasure the difference between the distributions of f(X,s) and s"X,s") for any 8 and in S. Following the literature (Chzhen et al. 2020: Gouic et al., 2020), we utilize the Wasserstein distance that conveys the distinction between probability measures by quantifying the expense associated with transforming one probability measure into another.
Definition 2 (Wasserstein-2 2 distance). Let μ and v be two probability measures on It with finite second moments. The squored Wasserstein-2 distance between μ and v is defined as
[
Wā(,
inf TE m. R x3*
liee 101111dy v)
(3)
where Fu is the set of probabilily mensures on R& x Ti such that ils marginal distribulions are equal to and v, i.e., for all y E T Tuw and all measurable sets A, C TR- it holds that 7(A x IR*) = =(A) and 7R* x vB)..
Suppose probability measures y and v both have density functions and the infimum 3 is obtained at y". There exists mapping Tu : R R such that 7 (Id, (,.,)) where # denotes the pushforward operator on measures and Id denotes the identity mapping (Santambrogio 2015). Here, Tu is known the optimal transport mapping from to v. With the Wasserstein distance, we characterize the geometric average of finitely many probability distributions by the Wasserstein-2 barycenter, which will be later used in the measure of unfairness considered in this paper.
Definition 3 For probability measures V2,..., Vm and PI,...,Pm Pm such that pt > 0, E Pt =1, the weighted Wasserstein-2 2 barycenter is given by
m
arg min v(v) = Σ Eamisonn
(4)
3.3 Post-processing with Distributional Parity Constraint
Let p. P(S = s) and Myjs be the probability distribution of f(X,S) conditioning on S = s for E S. Following Chzhen & Schreuder (2022), we then measure the unfairness of model f by the sum of the weighted distances between Vyis to their weighted Wasserstein-2 barycenters, namely,
u(f) min Σ Ep.Wa KES
As Wasserstein-2 distance serves as a distance metric on the space of probability distributions, W3( 0 and only if Vyls = V/Jl" (Kwegyir- Aggrey et al. 2023 Peyré et al. 2017). Hence, u(f) = 0 if and only if for any s E S, Vyjs', satisfying the distributional parity. However, when some level of unfairness is allowed,
only need to ensure U(J) S oU(()", where E [0. represents the tolerance of unfairness in terms of the fraction of the unfairness of f Like Chzhen & Schreuder (2022)
Awegvir-Aggrey et al. (2023), we consider the post-processing problem with distributional parity constraint as follows:
min R(J) s.t. u() S all(s))
/
where R(F) is defined in
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4 Structure of Optimal Solution
Problem 66] has been thoroughly studied by Chzhen et al. (2020): Chzhen & Schreuder (022) under the
setting that f is a singlo-output model. They provide an intuitive closed form of the optimal solution of ( for any a E [0, I] using the cumulative density function (CDF) of Vy for each 8. As they only focus on a single-output model, we expand their results to the multi-output case with some modifications in their proofs.
We present the extension of their results in this section and highlight the main difference.
When a 0 in G). the optimal solution of 16 can be characterized by the following theorem. Theorem Suppose Mg-ls has densily and finite second moments for each E S. Then
min R(F) = min Σ uifi=
= u(f")
(7)
.ES
Moreover, if fo and vy solve the first and second minimization in respectivcly, then vy is the distribution of fo and
fola,)) (f
where T 1s,00 R R is the optimal transport mapping from Vetl to vo.
By Definition 3 Po solving the second minimization in 7 is the barycenter of (pgla)es This suggests a post-processing method by transporting the output f" when S= ==s to that barycenter.
When k 1, Theorem is reduced to the structural results obtained by Chzhen et al. 20200: Gouic et al. (2000) (see, e.g., Theorem 2.3 in Chzhen et al. (2020)) Indeed, when k= ==1,
Ty (S" (2,
P. o Fy (x,, Σ
(9)
2'E8
where Fy-1s is the CDF of vg-ls and Qr-ss is the quantile function of Vy-ss i.e. Q5-1((() = inf E R (v) 2 f}.
In practice, when some level of unfairness acceptable, one can select a > 0 in 6 When a a= 1, f is the optimal solution. When a E (0, closed form of the optimal solution for is derived by Chzhen & Schreuder (2022); (wegyir-Aggrey et al. (2023). Although they are originally studied only under the single-output case, the same result holds for a multi-output model by almost the same proof. This closed form is presented in the following proposition, which motivates the way of trading off the error and fairness in our post-proccssing method in the next section.
Proposition 1 (Proposition 4.1 in Chzhen & Schreuder 2022) Suppose Vy-ls has density and finite second moments for each E S. For any 10, the optimal solution to denoted by fax satisfies (up to zero-measure sel)
(a(2,)) +) +1- (1-V)))(oz())
(10)
where fo is defined in Eq. 18.
Remark: It is not always possible to minimize R and U simultaneously by the same f. Therefore, model f is valuable as long as it achieves a value of R (or U) that cannot be further reduced without increasing Ut (or R). In fact, a funclion f(x,, 8) mensurable in x called Pareto efficient if there no function f' 8) measurable in such that one of the following two cases happens: (1) R(f) S R(f') and U(f) < U(f') (2) R(F) < R(f') and U()) U(("). The Pareto frontier is 2-dimensionol curve consisting of (U(f), R(f)) for all Pareto efficient f f's. Proposition 4.6 in Chzhen E Schreuder (2022) shows, the Pareto frontier for U and R is actually the curve ((f{)) with a varying from 0 to 1, where fa is from (10). Although d == in Chzhen & Schreuder 2022), their Proposition 4.6 holds for our setting also for any domension d 2 1 by the same proof.
5 Fair Post Pest-Processin with Finite Samples
Theorem I and Proposition I only characterize the optimal solution (6) when Vy-js has density. However, in practice, we only have access to finite set of outputs of f*; denoted by (f" se) where ((1,, si)

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is dataset sampled from the distribution of (X, S). p As result, we are not able to compute Vo and Is.0 exactly and apply (10) To address this issue when k = 1, plug-in principle is applied by replacing Ps- Fr-1..t (t), and Q 11 in (9 using their empirical approximation based on finite data sample (Chzhen et al.
2020; Gouic et al. 2020p- Because f" is a multi-output mapping in our case, (9) is not applicable. In the lollowing, we present how to employ the plug-in principle by approximating vo and T- in 8 by finite data sample. It consists of three main steps.
5.1 Optimal Transportation with Finite Samples
Cousider approximating vo and Ty by a collection of datasets D2 = {1o 8)] for s E S. We denote the empirical distribution of I on D, by Vy-(D.), Lee. M)-(D.) = 1 Σラ= (x),ss) for 8 E S, where 5 is the Dirac Tye
measure. Consider two discrete distributions in R*: p = - der and Di pey,y, where & E R*, E E R2 p." > 0, p > 0, Em= Pi = 1 and E S p = 1. As a special case of 3 the squared Wasserstein-2 distance between μ and V is
W = min YER
s.t.
Σ E11
p., VJ.
Σ = Pi.
(11)
1=1 1=1
where Gú =|5," Ej 12 and My represents the mass located & transported to Ej in order move distribution I to V. Also, can be viewed as a discrete distribution in R x R supported on (4', Eÿ)) for j = 1,.., By and = 1,,,.. ... Ny. Suppose 7" E R"=xR is the optimal solution of [1]. The optimal transportation from μ to v and from v to μe, denoted by T. and Tup respectively, are random mappings such that
N4 p
% py
P(T,,) (
((T) = (3)
(12)
for j= 1,.., nu and == 1, L,...,7y Ry.
With (2, given in 11 the barycenter of discrete distributions for i = 1...., m is also defined as the solution of Since ()-(D.) is the discrete approximation of Vy-ls: we propose to approximate s0 in Theorem by the barycenter of {Vy-ssssss with the weights p. = Exes n.' Unfortunately, Wasserstein
Although 4 can be formulated multi-marginal optimal transport problem (Agueh & Carlier 2011 and solved as linear program Anderes et al. 2016), its computational complexity scales exponentially in terms of m. To reduce the exponential computing complexity, instead of computing the barycenter exactly, we adopt the approach by Lindheim (2023 to construct its approximation. This approach achieves a good balance between runtime and approximation error in practice. We present this approach in the next section.
barycenters are NP-hard to compute (Altschuler & Boix- Adsera 2022
5.2 Approximate Barycenter with Finite Samples
The approach by Lindheim 2023 first computes the optimal transport mapping between Vy-(D.) to V)-(D;) for each pair of and s' in S, i.e., Top= satisfying 12 For simplicity of notation, denote TUMDSNRED by Ts.s. Then, they define a mapping (z))s P s, (ft, where Ps = n, n. (Evesn))) n.) and the expectation is taken over the random output of Te.s' following distribution in (1).. Here, (Mf*(af s)) is the weight average of the expected outcomes after transporting (*( to each of the [S] distributions. Finally, the approximate barycenter is constructed as a discrete distribution with n. (())())))))))))> Compared to the exact barycenter, this approximation requires solving [S]([S] - 1)/2 optimal transport mapping between two discrete distributions and thus has polynomial time complexity. This procedure is formally stated in Algorithum D
ET
E.Es n1 supports defined as follows io = E.es Σnn, Pe
Let m [S] and 1/ = Vr in Let Po be optimal solution of (4 ie, the barycenter of (Vy-(D.) It is shown by Lindheim 2023 that (Pb) (Po) < 2. This bound is significant because there is no
TThe technique we propose model-agnostic and can be applied directly the outputs (2 8)) does not require knowing {(2), $
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Algorithm 1 Approximate Barycenter
Input: A mapping I" Rd R& and a dataset. = {(ri, 8:)) ) sampled from the of (X,
Partition D into subsets based on and obtain D2 {(2; s) M for E E. for S.
3: Let Ps 2.'ss
4: for S S S do
5: Solve with μ = (D.) and V - V)-(D,,) to obtain Ta,s' = -oproseeoo
6: end for
7: Construct mapping Mff (=] 8)) = Σ E Ps ET, (f
8: Output: Discrete distribution in = Lees Ei=1 Ps (M(((Z):))))
polynomial tieer algorithm can achieve a ratio arbitrarily close to one with high probability Boix- 2022).
Altschuler &
Although Proposition is derived for continuous distribution, it motivates a heuristic post-processing method to update the output (xf, to
sa( Vaf" (2), (1 (1- Vf of (f2,,
(13)
for = 1,.... and E S, where a E [0. Note that T is not obtained during Algorithm 1 and needs to be solved separately. When a and (uf,s) is uniformly randomly sampled from D.: fn has the same distribution for any s, indicating the post-processed outputs satisfy distributional parity, However, (13 is only defined for existing data in D, for E S. In the next section, we propose method to extend this processing scheme to new samples from (X
5.3 Post-Process Out-of-Sample Data
Note that [a(4ff s) is defined for (af, 8) D, only because the optimal transport mapping T (o, 5. in
13 is
only defined on f s) with (x).s) D.. To extend the definition of (a 8) from D. to any (x, 8) E IR x XS,
we extrapolate (o over R using the Nadaraya- Watson kernel regression method (Nadaraya
1964).
Let K R + R+ be kernel function satisfying ((2) = K(-2)and Sgt K(z)dz = 1. Let h > 0 be a bandwidth. For any (I Ed x S, the kernel regression estimator of T, oo (f* (f (z()))) is
ne K
w))//) Tu 20 (f (z3, K K -f (2)3,) s)//h)
](,,)) Po (f,
(14)
Here, f(x, s) is the post-processed prediction for data s). Recall that Toy- is randomly mapping (see 12)). so is Tujo To analyze the fairness of f(x, 8) on the out-of-sample data distribution, we present the following proposition to characterize how much (X, violates the distributional parity after post-processing by bounding U(f ((f(X,)) and the Wasserstein distance between the distributions of f(X, on different groups. The proof is given in Scction A.2
Proposition 2 Suppose K(2) = 1 exp(-22). Conditioning on data D = ((r, si) it holds that
VIT
u(J(X 5 Σo
pon exp
(+))){
+I
Pe W (2 h
(15)
#ES
sES
Moreover, let Unl be the distribution of S(X conditioning on S = for E S. holds that
1 1111 W unr) SO (n² + n exp (Th² ) +0
(16)
In both upper bounds above, the first term is the error due to extrapolation which will converge to zero as the bandwidth approaches zero. The second term the error due the sampling error between training data D and the ground-truth data distribution. The second error is represented by the Wasserstein distribution
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Algorithm Post- Post-Processing Method by Transporting to Approximate Barycenter (TAB)
1: Input: A data point (z, E R² x S, a mapping f RR S + I2 and dataset D = { st) sampled from the distribution of S)
2: Partition into subsets based on and obtain D. = [( 8) 8) for 8 S.
Compute i/y by Algorithm I
4: for = 1,...,.. |8| do
Solve 11 with μ Vy-(D, (D.) and in to obtain T (py
6: end for
7: if (z, s) E D. then compute fa(z, s) as in 13 else compute fa as in 18
8: Output: fa(x,s)
( (D.)). As the data sizes n, for increase, the rate at which WT (Mgrer (D-())) converges zero has been well studied in literature. For example, according to case (3) of Theorem in Fournier & Guillin 2015), if there exists q 4 such that
Exx.s]js==ll((((xx511 < +0c, where {(x, is the expectation over (X,5) conditioning on = 8, then the Wasserstein distance in the second term in 115 and (16) satisfies the following concentration inequality
(V UF(.)) 2 n)) S a(n,. 1) + b(n,. for any E (0,1)
(17)
where
)xxp(-cn,))),
if 4,
a(n,, =
exp ( e (n/ log(2 2 1/n)))), 1/2),
if 4, if
and b(n,, 11)
2-
After extrapolating Tu, Po to T Tome, defined above, we can extend 13 for any out-of-sample data by updating (x) to
fa(2,)) = (z) va)s(z,s).
(18)
We then formally present this post-processing method in Algorithm 2 apply the original mapping in (13) instead of its extrapolation (18).
Note that when (x, 8) E Da, we still
Our post-processing method can also be extended to improve the fairness of model under the notion of distributionally equal opportunity distributionally equal odds introduced in the remark after Definition [ In particular, to improve the fairness in terms of distrrbbtionally equal odds, we can partition the training set D into | subsets {D")eey depending on the class label Y. Then we apply Algorithm
2 to each subset D to construct a separate post-processing mapping (x, like (14] for cach combination of group s and class y. To apply it on an out-of-sample data point, we first generate a predicted label ÿ using (x( s) and then convert sz) to fz(z, s) using the mapping constructed on D", The same procedure works for distributionally equal opportunity with respect to one class y E V, except that we only need to construct the mapping on D and post-process data for one class y.
6 Experiments
In this section, we apply the proposed post-processing method machine learning models with multiple outputs evaluate its offectiveness, including muntt--label/mmutiicche classification and represcntation lcarning.
Datasets. In experiments, include four datasets from various domains, including marketing do- main( Customer dataset 3 medical diagnosis( Chezpert Dataset (Irvin al. 2019)), face recognition ( (CelebA datasct (Liu et al.) 2015) and UTKFace datasct (Zhang & Qi 2017)). The details of these datasets are provided in Appendix
Baselines and Settings. To verify the effectiveness, we compare our method against six baseline approaches, including four post-procsssing methods: (1) al. (2023) (Hu et al. 2023), which introduces a post-

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processing approach for incorporating fairness into multi-task learning using multi-marginal Wasserstein barycenters; (2) Xian et al (2023) (Xian et al. 2023). which characterizes the inherent tradeoff of demographic parity in multi-class classification and proposes an effective post-processing method; (3)FRAPPE (Tifrea ct al. 2024), which proposes a generic framework that converts any regularized in-processing method into a post-processing approach; (4) Debiasing (Zhang et al.] 2018). : which presents a framework for miligating demographic group biases by introducing group-related variable and jointly learning a predictor and adversary, and two in-processing methods: (5)SimPPair (Liu et al. 2023), which focuses on training models for fairness-awan multi-label classification; FERM (Baharlouei et al. 2024 which presents unified stochastic optimization framework for fair empirical risk minimization regularized by f-divergence measures.
Specifically in our experiments, for baseline Adv Debiasing, both Predictor Block and Adversary Block are implemented by a two-layer neural network with 128 hidden units and tanh activating function, respectively, For baselines FRAPPE, SimFair, and Adv Debiasing, we train the models for 60 epochs with Adam optimizer, batch size as 64, and tune the learning rate in {1c-3, 1c-4]. We vary their weight parameter )(or a) in { 0.1, 1, 2, 4, 8, 10} to show the trade-off between classification performance and fairness. For J-FERM, we follow their paper to vary the weight parameter À in {0.1, 1, 10, 50, 100, 150} and tune the learning rate in {0.1, 0.01, 0.001} with their proposed optimization algorithm. For Hu et al. (2023) (Hu et al. 2023 we vary parameter a for their method in {0, 0.2, 0.4, 0.6, 0.8, 1.0). Following their paper, the parameter a for Xian et al. (023) are in {1, 0.16, 0.14, 0.12, 0.1, 0.08, 0.06, 0.04, 0.02, 0.01, 0.008, 0.006, 0.004, 0.002, 0.001, J.0. For our method, we experiment with a Gaussian kernel and h is chosen from {0.02, 0.04, 0.5, 13 based on input dimension, as smaller h theoretically and empirically leads to better performance but too small h may lead to numerical issues. We vary a for our method in (0, 0.2, 0.4, 0.6, 0.8, 1.0]. All the experiments are run five times with different scods.
6.1 Multi-label classification
For multi-label classification tasks, we experiment on CelebA dataset and Cheapert dataset. We compare our method with Siirair (Liu et al., 2023), FRAPPE (Tifrea et al. 2024), Adv Debiasing and the method Hu et al. (2023) (Hu et al. 2023), which essentially independently applies the post-processing method for a single-output model in Chzhen et al. (2020): : Chzhen & Schreuder (202)) to cach coordinate of the output of a multi-task classification model. Firstly, one ResNet50 (He et al. 2016) and one DenseNet121 (Huang el al., 2017) and are trained on the CelebA and Cherpert data respectively, then post-processing methods Hu et al. (2023) (Hu et al. 2023). ; Adv Debiasing and our proposed method are applied to the predicted probabilities of each task and methods SimFair (Liu et al. 2023) and FRAPPE (Tifrea et al., 2024) are applied to the extracted image features to train a new linear classification head. For both FRAPPE and SimFair, the fairness regularizer is defined as described in Equation 1 of the (Liu et al. 2023) paper, it has been demonstrated to be effective for promoting demographic parity. The results are summarized in Fig. Daa) and Lb). We can observe that, at the same level of accuracy, our method achieves much lower unfairness than baselines, which can be seen clearly when drawing horizontal lines across the figures. Particularly, on the Chexpert dataset, our method delivers fairer predictions with only a negligible decrease in predictive performance. From Fig. B(a) and b), we also note that learning- based basclines such as FRAPPE, SimFair, and Ade Debiasing tend to achieve lower unfairness but only at the cost of a substantial drop in accuracy, whereas processing- based observe that with the same fairness regularizer, the post-processing method FRAPPE achieves comparable fairness-error trade-offs to the in-processing technique SimFair, aligning with the findings reported in Tifrea et al. (2024).
methods like Hu et al. (2023) and our method maintain better balance. Furthermore,
6.2 Multi-class classification
To verify the effectiveness of our proposed method on multi-class classification tasks, we apply method to the customer datasct. A classic multi-class logistic regression model is built first on the training data and then post-processing methods are applied to the predicted probabilities of each class. For in-processing met the mulli-class logistic regression models are trained with fairness regularizer from scratch. We compare our method with FRAPPE (Tifrea et al. 2024), f-FERM (Baharlouci et al., (024)), Xian et al (2023) (Xian et al. 2023) and Adu Debiasing. Note that method Xian al (2023) directly produces the class labels instead of score function, so we can only perform the comparison on the classification accuracy and the unfairness based
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CelebA
CheXpert
Customer
0.825
0.90
0.820
Accuracy
0.08
0.86
Accuracy
0.810
0.815
Accuracy
0.30
0.84
Hu et 1202
0.00
Hu (2023)
0.26
IAB
0.82
Deblasing
0.800
0.24
Xian (2023)
Adu Deblasing
0.80
FRAPPE
FRAPPE
FRAFRE
0.78
SimRair
0,790
SinFaair
0.0
0.1
0.3
0.4
0.002 0.004 0.006 0,008 0,010 0.012 0.024
0.0
0.2
0.4
0.8
Unfaimess on Wasserstein Distance to Barycenter
Unfairness on Wasserstein Distance to Barycenter
Unfaimess on Predicted Classes
(a)
(b)
Figure 2: Multi-label classification on CelebA dataset (a) and Chexpert dataset (b); (c) Multi-class classifica- tion on Customer dataset.
Group
0332
6.1
Makruugg
2302
6.7
(203))
UnBomes Wasloste Deliance Barveedee
(b)
(d)
Figure 3: I-SNE visualization of representations on CelebA dataset. (a) Raw representations from SSL model; (b) Representations after post-processing(t = 0) by Hu et al. 2023): (c) Representations after
post-procesing(a- = 0) with TAB(Ours); (d) Performance on downstream tasks.
Rane
Rate
0.65
Hu (3123
355
sis th's zin
Unlamees Dinance aayestar
(a)
(b)
(d)
Figure 4: (-SNE visualization of representations UTKface Dataset. (a) Raw representations from CLIP
model; (b) Representations after post- post-processinga = 0) with Hu et al. (2023); (c) Representations after
post-processing(a= = 0) with TAB(O:::); (d) Performance on downstream tasks.
Equation which is in favor of their method. For FRAPPE, the fairness regularizer is defined as the Equation 2 proposed by Xian et al. 2023). We present the results in Fig, (c.) The figure demonstrates that our method effectively reduces unfairness measured by the metric proposed by Xian et al. (023) and achieves a superior balance between accuracy and unfairness compared to the baselines, particularly when a higher level fairness is required. Besides, we notice that although it is implied that methods F-FERM (Baharloue] ct al. 2024) and FRAPPE (Tifrca et al. 2024) are applicable for multi-class classification, their respective papers do not include experiments for this scenario, and their methods, in our experiments, fail to mitigate unfairness defined by Equation 2 even when it brings significant undesirable performance decrease.
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6.3 Representation learning
In this part, we explore the fair representation learning for self-supervised lear)ing(SST.) models and large pre-trained foundation models. We experiment CelebA dataset and UTKFace dataset. Due to the scarcity of the postprocessing methods for reprosentation learning in existing literature, we still compare our method with the post-processing method Hu et al. (2023) (Hu et al. 2023) and Adv Debiasing. For CelebA dataset, we first train an SSL model to learn representations with a dimension of 128, by employing the algorithm proposed in Yuan et al. (2022) on the whole training dataset. Then we apply post-processing methods to eliminate the sensitive information in the raw representations. Specifically, for Adv Debiasing, we utilize the raw representations as the labels for the Predictor as there is no task-specific labels in self-supervised learning. In other words, we aim for the model to remove sensitive information while altering the raw representation minimally possible. With processed representations we also build multi-label logistic regression model to evaluate the performance on downstream tasks. For UTKFace dataset, raw representations are generated by a relcased CLIP (BiT-B/// model (Radford ct al. 2021 which is pro-trained on 400M text-image pairs, with a dimension of 512.
We evaluate the unfairness of the representations as well as the accuracy performance of the downstream tasks. The results summarized in Fig. 3 and 4 By examining the representations of Fig. 3(a) and 4 a), it's evident that both the SSL model and CLIP model disclose sensitive attributes prominently. Even after applying post-processing techniques proposed in Hu et al. (2023) (Hu et al. 2023), this exposure persists, depicted in Fig. B(b) and Fig. [b). However, through our proposed methods illustrated in Fig. 3(c) and Fig. [c), we are able to achieve fairer representations with respect to sensitive groups. Notably, from Fig.
and ddd,, we can observe that our method achieves a better tradeoff between downstream performance and distributional parity, whereas Hu et al. (2023) (Hu et al., (023) fails to improve the distributional parity because, as discussed in Fig. [ the elimination of unfairness in individual outputs may not necessarily mitigate unfairness in the joint distribution of outputs.
7 Conclusion and Discussion
In this paper, have proposed post-processing method to enhance fairness for multi-output machine learning models, which is underexplored in the literature. Our approach employs optimal transport to move a model's outputs across different groups towards their empirical Wasserstein barycenter to achieve the model's distributional parity. We have developed an approximation technique to reduce the complexity of computing the exact barycenter and a kernel regression method for extending this process to out-of-sample data. Extensive experimental results on multi-labelmmulticcaasss multi-class classification and representation learning demonstrate the effectiveness of our method.
One limitation of this work is that the notion of fairness in Definition I we pursue is strong while weak notion such as (2) might be sufficient for some specific applications. To achieve a stronger sense of fairness may lead to more decreases in the predictive performance than targeting weak sense. Therefore, applicability of the proposed hod may vary depending on the specific use case. Additionally, the lack of theoretical convergence analysis of the proposed method as the training sample size increases is another limitation, which is an important future work.
Broader Impact Statement
This paper is to attain distributional fairness in machine learning-driven decision-making, particularly concerning various demographic groups such as gender, age, and race. When enforcing group fairness, it may potentially disadvantage certain individuals. Other than this, none we feel must be specifically highlighted hcre.
Acknowledgments
We thank anonymous reviewers for constructive comments. G. Li, Q. Lin, and T. Yang were partially supported by NSF Award 2147253.
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Yao Yao, Qihang Lin, and Tianbao Yang. Stochastic methods for optimization subject to auc-based fairness constraints. In International Conference on Artificial Intelligence and Stalistics, pp. 10324-10342 PMLR, 2023.
Zhuoning Yuan, Yuexin Wu, Zi-Hao Qiu, Xianzhi Du, Lijun Zhang, Denny Zhou, and Tianbao Yang. Provable stochastic optimization for global contrastive learning: Small batch does not harm performance. In International Conference on Machine Learning, pp. 25760-25782. PMLR, 2022.
Xianli Zeng, Edgar Dobriban, and Guang Cheng. Baycs-optimal classifiers under group fairness. arXiv preprinl ar Xio:2202.09724, 2022.
Brian Hu Zhang, Blake Lemoine, and Margarel Mitchell. Mitigating unwanted biases with adversarial learning. In Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society, pp. 335 340, 2018.
Fengda Zhang, Kun Kuang, Long Chen, Yuxuan Liu, Chao Wu, and Jun Xiao. Fairness-aware contrastive learning with partially annotated sensitive attributes. In The Eleventh International Conference on Learning Representations, 2022.
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Published in Transactions on Machine Learning Research (03/2025)
A Proofs
Before providing the proof of Theorem let's review some additional results about Optimal Transport theory.
The next result shows that as long as two measures admit a density with finite second moments, there exists a unique deterministic optimal transport map between them.
Lemma (Theorem 1.22 in Santambrogio (2015)) Let μr, v be two measures on RRt with finile second moments such that μ has a density and let X p. Then there erists a unique deterministic mapping T R R such that
= - that (X,T(X)) ~ Fw where y an optimal coupling.
A.1 Proof of Theorem
This proof originally from the proof Theorem 2.3 in Chzhen et al. 2020). We extend their results from single output case to the multi-output case with some modifications in their proofs.
Theorem Suppose Uy has density and finite second moments for each E E. Then
min R() -u() min Σ Ep.Waleraa))) uif=0
Moreover, if fo and My solve the first and second minimization fo and
respectively, then vo the density of
fo(2, =
(f" (f"
where Tyloo : TR. - & is the optimal transport mapping from V/- to vo- Proof First, we'r 'd like to show that
min BlIf f(X, 2 min Σ (2 u(T=0
sES
Let g. : Rd x + R& be the minimizer of the l.h.s of the above equation and denoted by Va the distribution of g. Since Vy admits density, with Lemma 1 for each s E S there exists T/ such that
Ep.W ug) = Eps Ps / (rinna
Σ
(1f" - Ty- (f (x. s))1 ? dP (x)
Ps
JRd
P.E [IIf 8)
=s]
= EX 5) - ((X,S)\'
where we define g(I,8) = -Ty- (f (1) for all (a. E Rdd x S. With optimal transportation, g(X,8)|S = follow the distribution Vn for any s E Then we have
u(g) = min Σ (P) =0 SES
which indicates ğ is fair. By optimality of g we have
(XI g(X S)1l2 (XI 3(X,
Due to definition of W2 for each s we have
W3(P S EXI 9(X
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Published in Transactions on Machine Learning Research (03/2025)
Then con conclude
min - Σ P.
uif=0
This implies that
min 4())-
min Ep.Wg(ur-aa)))
(19)
SE8
Second, we are going show that the opposite inequality also holds. To this end, we define vo as
vo E arg min Σ
ps P.
SES
Since we assume Vf-ls admits density, with Lemma
there erists Tygajao as optimal transport map from
Vgna to Vo- And we define fo for all (x,, 8) E Rd x S as
fu( Turrunn o fo(
By the definition of fo and the Lemma 1 we have
min (V - (X) fo(x
(20)
SES
Moreover since vo is independent from S, using simiilar argument above can show that fo is fair, and it yields
min Σ po 1
min -X uff)=
(21)
SES
Therefore, combining Eq. 19 and Eq.
21
we showed that
min Σ
min ps W2
(X S(X,
u(f)=
SES
Thanks to Eq. 20 we can also have
( S(X
and since fo is fair can put g for This proof is concluded.
A.2 Proof of Proposition
Proposition Suppose K(2) 22n exp(-22).
Conditioning on data D {(x,, 8:)]
holds that
u(f(x,)) < Σ
ponn exp
(-1²)) Σ o
112 (-7²)
(-1
+
Pe C
sES
xES
Proof Recall that (X is the ground- distribution. Let
f(x, s) = T Toy (f (x, 8)).
Let Vi be the distribution of S), and via be the distribution of f(X, S) conditioning on S = s for E S. Suppose Vy has no supports, denoted by th c RR* In the entire proof, we assume D, is already sampled and thus deterministic for S.
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Published in Transactions on Machine Learning Research (03/2025)
By 5
we have
u(f) = min Σ
W v) Σ (v
8ES
#ES
(22)
1
2p. W (/(D.)
sES
SES
where the second inequalily is the briangle inequalily. Nerl, we bound W (u F(D.)) and W ( vn) from above separately.
Consider any s E S. We denote the discrete distribution of f(X, S) when (X,S) is uniformly sampled from Da as "f(D.) Let v(.) I& + be any L-Lipschila continuuous function on IR+, meuning that (Ø(E) - -(E') l|5
El'll for any and & in R We have
S/L ((E)dU = The
1
n,
ES (2f and f @(E)din =
R2 The Σ
Consider any i (I,...,,,, ma). Let E, be the conditional erpectation conditioning on the outcome of random mapping To To (f (x), By 114h and the 1-Lipschitz continuity of 6(.), we have
Eo (4) ET(Tor (ff,
E ETd(T (e (z), (I (S (2), (1))
Ei K (1) (4), s))/k) EE, Po (s" (47
Po
Σ/ K (()(x) 8) (1),
Σe= K (1f))))))) - (z(), s)/hh) RE, (Ty (f" (x), To To (f
ΣØ K ((f (z),
A ().,DD))mi
<n, exp
mpg.max
2h2
where
(p.))
min me.
()) - (2z, 8) and Avo. = max
[|& - E,
=1.==....,n
As result, we have
(D.))..,mnn
((E)duF(D)) - Φ(E)d I cxp
As
2h2
for any & that is 1-Lipschitz continuous function on R
By Kantorovich-
Rubiisseenn duality, we have
( 80) S n exp
A
(D.),,ii
A 2
O (12 exp (- h22 (-12)-
(-1)
(23)
h
:*+) +R be any 1-Lipschitz continuous function on R* Let ET be the expectation over the random mappings T, Tupr:,,) (x; (x;,s) 8)) for l.....ng. ng. We have
Again, let o()): R*
J (E)d / Er and [ o(E)dun= - (E)
where
Ene K (5-f"((((o))
Teyreo (M7,,, ())
((E)
(5-1((27,0)
E K
h
20

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Published in Transactions on Machine Learning Research (03/2025)
is random value function whose randomness is from To (o.) (xf, for 1,...,, ng. Since K is ( continuous and & 1-Lipschitz continuous, (({) is O R2 )==) continuous in E given any outcomes of random mappings To (12, for i 1,..,, ng. Therefore, E, (() is also -Lipsshht continuous.
By Kanlorovich- Rubinslein iiubonooic--Ruubinsleein
inequality (Edwards 2011
there exists a consland C such that
J o(E)du / ((E)du f (r /
C BTD(E)duM1 &T (a(
which implies
C2 W ((n,), S Wi(UFIa ht
(24)
Applying
23 and 24 to 22 gives the first conclusion.
Consider any two groups and in S. Applying 23 and 24
and the triangle inequality again, we have
(F) I Wy Do) + W (v).
SWa "I(()) + W ((D)) 200 + Wā(un) Bb) + W
<0 (ng n².) exp (T²²) + +0 (111)10011100))
which gives the second conclusion.
B Details of Datasets
In this section, we provide more details on the datasets we used in the numerical experiments. We include four datasets from various domains, including marketing domaing Customer dataset [), medical diagnosis( Cherpert Dataset (Trvin et al. 2019h), face recognition (CelebA dataset (Liu et al. 2015) and UTKFace.
dataset (Zhang & Qi 2017)).
The Customer dataset has 8068 training samples and 2627 testing samples and the task is to classify customers into anonymous customer categories for target marketing. We partition the data into four sensitive groups based on the gender and the marital status of customers: married female, unmarried female, married male, and unmarried male.
The Cherpert dataset contains 224,316 training instances and the task is to detect five chest and lung discases based on X-ray images. Due to the high computational complexity of solving optimal transportation between large datasets, we sample 5% instances from the original training data as the training set and sample another 5% as the testing set. The CelebA dataset contains 162,770 training instances and 39,829 testing instances and the task is detect ten attributes (chosen based on Ramaswamy et al. (2021)) of the person in an image, which are being attractive, having bags under eye, having black hair, having bangs, wearing glasses, having high check bones, being smiling, wearing hat, having a slightly open mouth, and have a pointy nose. For the same computational reason, we sample 5% instances from the original training data the training set and sample 20% from the original testing data as the testing set. For both Chexpert and CelebA datasets, we partition the data into four sensitive groups based on gender and age: young female, old female, young male, and old male. UTKFace datasct consists of 23705 face images with five groups in terms of raco(i.c. White, Black, Asian, Indian, and Others) and we randomly split into training and testing (8:2) sets. And the task is classify gender and age (25 S age I 60, customized by us) based on face images. All the data in UTKFace datasct are utilized.
21


Reviews:
[{"review_id": "IPDJY7kCmB", "reviewer": "Reviewer_NBty", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1) The paper is written in a clear way. The algorithm and the provided theory is technically correct.\n\n\nWeaknesses:\n\n1) The generalization to the multiple classes does not seem a challenging issue in the first place. Please consider to explain why it is not straightforward given [1].\n\n2) There are several method in the literature (both in-processing and post-processing) that can handle multiple sensitive attributes. In particular, [2], [3], and [4] propose their methods in a general case where both sensitive attributes and labels can be non-binary. Please consider comparing your approach with other in-processing (optional) and post-processing (mandatory) methods that can handle multiple labels/classes. \n\n3) In paper [1], Section 7, they propose a post-processing fair approach and they present an approximator of the Barycenter for a given $n$ iid samples. How your estimator is different from the one proposed in that paper? It is important to discuss in what scenarios their estimator fails and how your estimator can address those scenarios.\n\n4) The presented fairness measure only works for demographic parity notion. Is that possible to generalize it to other popular notions such as equalized odds and equal opportunity (by conditioning the joint distributions on $y = 1$)? \n\n5) In the experiments (Figure 2 in particular), the fairness is measured by Wasserstein Distance to Barycenter, which is a measure proposed and directly optimized by the paper. A more fair comparison is to use demographic parity violation (the true unfairness measure of interest) as the fairness measure. Besides, unlike the demographic parity violation, it is not clear how good/bad is Wasserstein Distance to Barycenter = 0.2 for instance. Therefore, please consider to measure fairness with more general criteria such as $p$% rule and demographic parity violation.\n\n\n[2] Baharlouei, Sina, Shivam Patel, and Meisam Razaviyayn. \"f-FERM: A Scalable Framework for Robust Fair Empirical Risk Minimization.\" In The Twelfth International Conference on Learning Representations.\n\n[3] Tifrea, Alexandru, Preethi Lahoti, Ben Packer, Yoni Halpern, Ahmad Beirami, and Flavien Prost. \"FRAPPÉ: A Group Fairness Framework for Post-Processing Everything.\" In Forty-first International Conference on Machine Learning.\n\n[4] Liu, Tianci, Haoyu Wang, Yaqing Wang, Xiaoqian Wang, Lu Su, and Jing Gao. \"SimFair: a unified framework for fairness-aware multi-label classification.\" In Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 12, pp. 14338-14346. 2023.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a fair supervised learning framework for multi output/multi class problems under the statistical parity notion. To do so, the paper considers the  weighted distance of every joint distribution of predictions and sensitive attributes to their common barycenter w.r.t. the Wasserstein-2 distance. The main contribution is to generalize the formulation proposed in [1] to multiple classes and multi-tasking problems. Furthermore, due to not accessing to the true joint distributions of predictions and sensitive attributes, they come up with an approximation of Barycenter based on $n$ given iid samples. The experiments demonstrate the superiority of the proposed  approach over several baselines in terms of fairness-accuracy tradeoffs on different multi class/multi label datasets.\n\n\n\n[1] Evgenii Chzhen and Nicolas Schreuder. A minimax framework for quantifying risk-fairness trade-off in regression. The Annals of Statistics, 50(4):2416–2442, 2022.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the Weaknesses section in the previous section.", "broader_impact_concerns": "No Concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Qg9i0lPbxQ", "reviewer": "Reviewer_F5TY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. Extends fairness from single-output models to multi-output settings, addressing an important gap in the literature. \n\n2. Applicable to diverse tasks and models without retraining, offering flexibility and computational efficiency. \n\n**Weaknesses**\n\n1. The method focuses on only one fairness measure, and it is a strong fairness notion. This means that there are cases where the focus on achieving strong distributional parity may lead to large drops in predictive performance. \n\n2. There are limited baselines because the authors choose to primarily target post-processing methods. It would be helpful to have insights on how it compares against pre and in-processing methods.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a post-processing method for achieving fairness in multi-output machine learning models. It focuses on enhancing distributional parity, which is a task-agnostic measure. The method uses optimal transport to align model outputs across different groups with their empirical Wasserstein barycenter, while maintaining predictive performance. Experimental results on multi-labal/multi-class classification and representation learning tasks demonstrate its effectiveness compared to baseline methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The work would be significantly strengthened by incorporating a broader range of baseline methods for comparison. While the authors compare their approach to existing post-processing methods and one in-processing method, the inclusion of additional baselines would provide a more comprehensive evaluation of the proposed method's relative strengths and weaknesses. This would also help clarify the scenarios where the method offers the most advantage.", "broader_impact_concerns": "There are no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "zBKv2ilvCR", "reviewer": "Reviewer_DeBp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The method generalizes fairness notions like Statistical Parity from single-output to **multi-output problems**. To my best knowledge, the motivation is strong since there is a lack of papers discussing multi-output problems in the fair ML community, especially **post-processing**.\n- The paper makes **a good combination of previous work to solve multi-output problems**, such as \n  - post-processing with fairness constraint defined through optimal transport, \n  - the optimal solution as well as its proof technique,\n  - the approximation to compute the optimal transport, \n  - and the Nadaraya-Watson kernel regression method.\n- The numerical experiments demonstrate the method’s superiority compared to existing post-processing techniques, validating its practical utility.\n\nWeaknesses:\n- Since the main problem setting, definition, solution, proof and algorithms are strongly motivated by previous work (as the authors state), I would suggest the authors to incorporate more analysis, which are interesting and already presented in these papers. In particular,\n  - The fairness guarantees from Proposition 4.1 in [1] and Theorem 7.2 in [2].\n  - The analysis of Pareto frontiers in [2]. \n- In the experimental parts, the paper states that: \"Due to the limited post-processing methods in the literature for multi-output problems, we also incorporate an in-processing method, Adversarial Debiasing (Adv Debiasing) (Zhang et al., 2018), which can be **extended** for postprocessing.\" However, you may not need to \"extend\" Adversarial Debiasing since both **post-processing** and **multi-output problems** are important advantages that make your paper different from others. Therefore, you can compare your algorithm with **some pre/in-processing methods for multi-output problems** (such as [3-4]) to make your paper more influencial.\n- Equalized odds and equal opportunity [5] is also a popular fairness definition in this community. The researchers may naturally raise the question: Can this algorithm be extended to equalized odds and equal opportunity? The authors are encouraged to discuss this fairness definition regarding the proposed algorithm and theoretical analysis.\n\n[1] Fair Regression with Wasserstein Barycenters. NeurIPS 2020.\n\n[2] A minimax framework for quantifying risk-fairness trade-off in regression. The Annals of Statistics 2022.\n\n[3] SimFair: A Unified Framework for Fairness-Aware Multi-Label Classification. AAAI 2023.\n\n[4] Beyond Adult and COMPAS: Fairness in Multi-Class Prediction. arXiv preprint arXiv:2206.07801, 2022.\n\n[5] Equality of Opportunity in Supervised Learning. NeurIPS 2016.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a post-processing method to enhance fairness in **multi-output** machine learning models, addressing limitations of existing methods that are typically designed for single-output models and task-specific fairness measures. The proposed approach focuses on achieving distributional parity, a task-agnostic fairness metric, using optimal transport mapping to align model outputs across different groups towards their empirical Wasserstein barycenter. To manage computational complexity, an approximation technique is used for the barycenter computation, and a kernel regression method is introduced to handle out-of-sample data. Empirical evaluations demonstrate the method’s effectiveness in improving fairness on **multi-task/multi-class** classification and representation learning tasks, outperforming current post-processing baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "For theoretical part:\n- The fairness guarantee of an algorithm is beneficial and will make your paper more theoretically sound for the fairness community. Note that this paper is motivated by [1-2], and both of them present fairness guarantee. Therefore, I suggest the authors to incorporate a bound for their fairness definition, i.e., \"Multi-output Distributional Parity.\" If the authors cannot derive the bound, it is also recommended to add a discussion regarding the fairness guarantee.\n- Please discuss equalized odds and equal opportunity in the context of this paper.\n\nFor empirical part:\n- I believe that it is important to investigate the fairness-utility trade-off [3], especially in the context of multi-output problems. Therefore, I suggest the authors to incorporate some analysis regarding Pareto frontiers, like Figure 4 in [2], which is considered an important reference of this paper.\n- Please make a comparison with other fairness-aware algorithms for multi-output problems. See the Weaknesses part.\n\n[1] Fair Regression with Wasserstein Barycenters. NeurIPS 2020.\n\n[2] A minimax framework for quantifying risk-fairness trade-off in regression. The Annals of Statistics 2022. \n\n[3] FACT: A Diagnostic for Group Fairness Trade-offs. ICML 2020.", "broader_impact_concerns": "The authors may add a section: \"Broader Impact Statement\"", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a post-processing method to enhance distributional parity, a task-agnostic fairness measure, in multi-output machine learning models. Extending previous fairness research on single-output models, this work leverages optimal transport mapping to align the model’s output across different groups. Reviewers agree that this work is well-motivated (DeBp), technically solid (NBty), the proposed post-processing method avoids retraining and is flexible (F5TY), and it fills an important gap between single-output and multi-output fairness in machine learning models (F5TY). Meanwhile, reviewers also raise several concerns regarding the contribution in comparison with previous works (NBty, DeBp), model prediction performance (F5TY), additional criteria (NBty), and additional baseline comparisons (DeBp, F5TY). The authors make a good effort in the rebuttal phase, and reviewers agree that the concerns are resolved in the revised version. While reviewer DeBp highlights that the gap between previous single-output problem and the multi-output problem in this work might be small, reviewer DeBp acknowledges that this is a solid study on a previously under-explored problem.

Given the consensus of reviewers, the AE recommends the acceptance of this paper.
Decision: Accept (poster)