Optimal Transport of Classifiers to Fairness
Authors: Maarten Buyl
Tijl De Bie
Venue: NeurIPS 2022 Accept (2022)
Keywords: fairness
optimal transport
projection
regularization
classification

Abstract: In past work on fairness in machine learning, the focus has been on forcing the prediction of classifiers to have similar statistical properties for people of different demographics. To reduce the violation of these properties, fairness methods usually simply rescale the classifier scores, ignoring similarities and dissimilarities between members of different groups. Yet, we hypothesize that such information is relevant in quantifying the unfairness of a given classifier. To validate this hypothesis, we introduce Optimal Transport to Fairness (OTF), a method that quantifies the violation of fairness constraints as the smallest Optimal Transport cost between a probabilistic classifier and any score function that satisfies these constraints. For a flexible class of linear fairness constraints, we construct a practical way to compute OTF as a differentiable fairness regularizer that can be added to any standard classification setting. Experiments show that OTF can be used to achieve an improved trade-off between predictive power and fairness.

Optimal Transport of Classifiers to Fairness
Maarten Buyl
Tijl De Bie Ghent University
Ghent University maarten. eayy@tteent.o. be
#ijl.debbe@ugeet. be
Abstract
In past work on faimess in machine learning, the focus has been forcing the prediction of classifiers to have similar statistical properties for people of different demographics. To reduce the violation of these properties, fairness methods usually simply rescale the classifier scores, ignoring similarities and dissimilarities between members of different groups. Yet, hypothesize that such information is relevant in quantifying the unfairness of a given classifier. To validate this hypothesis, we introduce Optimal Transport Fairness (OTF), a method that quantifies the violation of fairness constraints as the smallest Optimal Transport cost between probabilistic classifier and any score function that satisfies these constraints. For flexible class of linear fairness constraints, we construct practical way to compute OTF as a differentiable fairness regularizer that can be added to any standard classification setting. Experiments show that OTF can be used to achieve an improved trade-off between predictive power and fairness.
Introduction
Machine learning methods are increasingly being deployed for automated decision making due to the potential benefits in terms of efficiency and effectiveness. Yet, making automated decisions about people comes with substantial legal and ethical risks, as evidenced by more and more cases of undesirable algorithmic discrimination [ 13 10] with respect to the protected traits of individuals. Even decisions that appear neutral because they do not use such sensitive information, may still disproportionately affect certain groups indirectly (38
The research field of fairness in machine learning 30
therefore studies ways which model's discrimination with respect to person's sensitive features be reduced removed. Many notions of fairness have been proposed for binary classification tasks (37]. A popular example is demographic constraint that enforces equality between the rates which positive
parity, which is expressed decisions are made for each protected group 6
Motivation For a given fairness notion, we would like to tune a model that satisfies the constraint that expresses the notion. To this end, it can be practical to quantify the violation of this constraint as a fairness regularization term that can be added to any probabilistic classifier (26). Such an approach provides an incentive the model learn parameters that result in fair probability scores, e.g. by learning to ignore highly biased features. A straightforward fairness regularizer the norm of the difference between the quantities that should be equal according to the faimess notion 31]
However, such a quantification of unfairness may be limiting, as it only considers the statistical properties of the model's 's output, while ignoring the input features for which those scores were computed. We argue that, to quantify the unfairness of model as a fairness regularizer, it can be beneficial to more strongly consider data points with similar features but significant unfair discrepancies probability scores. To this end, we employ Optimal Transport (OT) 33 way to measure these discrepancies while taking feature similarity into account.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Contributions
We propose to quantify the unfairness of probabilistic classifier as the Optimal Trunsport to Fairness (OTF) cost, which is defined as the smallest OT cost between the score function of the classifier and any fair score function over the same data.
We make novel derivations compute OTF as a differentiable fairness regularizer that can be added to the training loss of any probabilistic classifier and is efficiently computed for popular notions of fairesss such as demographic parity and equalised odds. Our approach is capable of handling multiple sensitive variables, that can be categorical or continuous.
In experiments, our method shows its benefit of increased flexibility over other OT methods. In some cases, it also achieves trade-off between predictive power and fairness that is significantly more effective for equalized odds.
2 Background
2.1 Fair Classification
Let Z X x {0, 1 denote the sample space, from which we draw samples Z S. and try estimate the output label Y E {0, 1} from input features X X, without discriminating with respect to sensitive features S E S. It is assumed that X' Rdx and S C Rs We sample binary predictions Ý E {0, 1} from a probabilistic classifier h X [0, 1] that assigns score l(X) to the belief that a sample with features X belongs to the positive class. Our general goal in fair classification is to minimize loss function Ly (h), in our case the cross-entropy between h(X) and Y. while also scoring well on a fairness measurement of h(X).
2.1.1 Group Fairness
The fairness of classifiers is commonly enforced through some notion of independence from sensitive information S E S. In particular, group fairness is concerned with constraints over statistical measurements of this independence over all members of protected group 30). For now, assume that S consists of single, categorical feature with ds values. We denote S E S as the one-hot encoding with values S {0, 1}. This encoding is indexed by values from [ds] [0, .O., ds 1}.
Assuming that the predictions Y are randomly sampled from a probabilistic classifier h(X), then the traditional fairness notion of demographic parity (DP) is equivalent to enforcing zero covariance between h(X) and each group in S, which we refer to as Probabilistic Demographic Parity (PDP):
(PDP)
wk E [ds] Ez (h Ez (x() Ez 0.
Ez
(1)
Remark 1. If Y is not sampled from I(X) but instead decided by a threshold (e.g. Ý= 1 h(X)= > 0.5), then Eq. (I) is a relaxation of the actual DP notion and may therefore admit unfair models (29). However, such threshold introduces a discontinuity with respect to the input, making it difficult efficiently find fair classifiers practice (36]]] I. Moreover, by considering the probabilities directly, we account for the uncertainty expressed by the probabilistic classifier.
The notation in Eq. T is easily extended fairness notions that condition on other variables. For example, the equalized odds (EO) (21] fairness notion conditions DP on the actual label Y. We apply EO to probabilistic classifiers by conditioning the covariance constraint of PDP on the one-hot encoding of Y and retrieve the following expression for Probabilistic Equalized Odds (PEO):
(PEO) vl (0. 1), Ak E [ds] Ez (M(X)S.Yi
Ez (X))) Ez [S = 0.
2.1.2 Linear Fairness
The PDP and PEO covariance constraints in Eq. ( and Eq. written as linear sums over the probabilities given by the classitie h. As was done in past work []]. we are generally interested in all fairness notions that can be enforced through such linear constraints.
2 are practical, because they can be
For example, if S only contains two categories, then denote S {[1,0), [0, 1]} with |S] ds =2.

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Table Mapping of binary fairness notions onto the g functions of their probabilistic version be written as linear faimess notions.
Predictions Constraint
Linear Fairness Notion
(P)DP
P(Y = S) = P(Y =)
gi:
(P)EO
P(Y P(==1|S,)) P(Y = 1
gk+tid Vi
St
E2
Definition (Linear fairness notion). notion of fairness is linear fairness notion when the set F of all score functions X + [0, 1] that satisfy it is given by
F : X 00. 1] Ez (g (X)((x) Odz
(3)
with Oæ a vector of dy zeros and the vector-valued function g(Z): Z - Rd constant w.r.t f(X).
PDP and PEO are clearly linear fairness notions, with their g function listed in Table Linear constraints are practical because the set of fair score functions F is convex. Moreover, multiple linear fairness notions can be combined into a new linear fairness notion by concatenating their constraint vectors. Sensitive information that is inherently continuous, like age, can be considered as a single dimension in S, thereby allowing for mix of categorical and continuous sensitive attributes.
However, under linear fairmess, non-linear dependence between score function f(X) and continu- ous sensitive attributes S can remain undetected. Also, some well-known fairness notions are not linear. For example, notions related to sufficiency involve conditioning on f(X). meaning that fXX) also shows up in the function. To this end, approaches have been proposed [9] that can approximate classifiers with non-linear faimess through a reduction to problems with linear constraints.
2.2 Optimal Transport of Score Functions
Optimal Transport (OT) 32 theory considers the problem of moving mass from one measure to another at the smallest possible total cost. Here, we let every score function f' correspond with the measure f defined over the imput space X' endowed with the Borel o-algebra: 0, & EEED f(=))2. with dR the Dirac measure and Dx the input features of samples in a collection D. Note that the input space measure of is not normalized, though this is not necessary to apply OT theory. In what follows, we implicitly consider the score functions h and f as their corresponding input space measures 0, and , when used in the OT problem. See the Appendix C.1 for further clarification.
For collection D of n samples (e-g. full dataset or only batch from it), let h and f denote the n-dimensional vectors of score function values for all data points, i.e. h; h((,)) and f; f(;;). Furthermore, for non-negative transport cost function defined over X' X, let C E RTx" represent the matrix cost terms, i.c. C, c(2,, 22). Similarly, with (I,, 2zy) the coupling that reflects how much score mass was transported from I; to zy, define the matrix P E R with P.ş. = T (2,, 2y). The OT cost is then simplified to
OT(h, f) min (C,P) PeIT(h.))
with
[[h (P R Pl, h, -h, =f]
where 1, is the n--dimensional vector of ones and (C,P)=
= Eii C.p..s.
3 Optimal Transport to Fairness
Assume that a set Fi is available that denotes all score functions which satisfy the required notion of fairness. We would like to quantify the unfairness of a classifier's 's score function h as the amount of 'work" minimally required fair projection onto F, i.e. the f E F that is closest to h. We measure this closeness as the OT cost between h and f, as we then assign a higher unfairness cost to h if scores need be transported between highly dissimilar individuals in order to reach fair function f F.
make a member of this set by measuring how far h is from its
3

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Thus, we propose to quantify unfaireess as the Optimal Transport to Fairness (OTF) cost, i.e. the cost of the OT-based projection of h onto F:
OTF(h) min Oh, fEr
(5)
In what follows, we expand on the OTF method by constructing in three steps.
1. In Sec. 3.1 we directly express the (h) objective as linear programming problem by making the assumption that F is defined by a linear fairness notion as in Def.
2. In Sec. 3.2
we add entropic smoothing to OTF(h), thereby making the resulting OTF,
cost differentiable fairness regularizer that be computed efficiently.
3. In Sec. 3.3
address the fact that due this smoothing, OTF, (h) does not necessarily equal zero for a fair h. To this end, we subtract the adjustment term OTFR,(h) resulting in the adjusted OTF (h) cost.
Finally in Sec. 3.4 we show how OTF?(() can be minimized w.r.t. (the parameters of) h.
For all derivations and proofs, we refer to Appendix A.
3.1 Optimal Transport to Linear Fairness
To compute the minimization in Eq. ( first that in the OT f) cost in Eq. (4,, f only shows up in the constraint 1²1, on the column marginals of coupling matrix P. It therefore suffices to weaken this constraint 1, E F. Next, recall that linear fairness notion enforced through vector of constraints on the expectation of g( f(X).. For a collection D of n samples, let Gej = (zy)), i.e. E d the constraints matrix with a row for every constraint and a column for every data point. It defines the linear fairness notion for D.
Definition 2 (OTF). For linear faimesss notion expressed through constraints matrix G for samples and with non-negative cost matrix C, the Optimal Transport to Fairness cost for score function h X [0, 1] is computed
OTF(h) min (C,P)
PEIL (h)
with
11 (h) (P E RRX" PL, h, GP =0
The optimal coupling computed for OTF(h) implicitly transports the scores of h to the fair vector f given the coupling's 's column marginals (i.e. f= pf 1,,) Note that for any coupling matrix that satisfies the first constraint (P1,, h), we have that h h² 1,, 1 meaning that and this implicitly found vector sum to the same value. Scores of h are therefore only transported fair score function and no extra scores are created or destroyed.
For Púj 2 O, the optimization problem in Eq. 6 can be solved through linear programming.
However, its solution is not differentiable with respect the score function h 112 17 It is thus impractical as a fairness regularization term.
3.2
Entropic Smoothing
A well-known trick to make OT costs differentiable is to apply entropic regularization smoothing
which is done by maximizing the entropy of the coupling while optimizing the OT cost. We apply the same principle to make OTF differentiable.
12 33
Definition 3 (OTF.). For linear faimess notion expressed through constraints matrix G for samples and with non-negative cost matrix C, the smooth OTF cost with smoothing strength e for score function h X' + (0, 1] is computed as
OTF, (h)
min (h)
with H(P) = regulates the strength of entropic smoothing.
Púj (log (Pt)) 1) the entropy of coupling P and hyperparameter that

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The use of an entropy term in Eq. 7 requires some justification, since we do not assume that h represents normalized probability distribution (i.e. sums to one). Therefore. P also does not represent normalized joint distribution. However, the H()) term still provides the practical advantage that it only admits couplings with P, O, which is why we require that h, Furthermore, the addition of H H(P) makes the objective cost strongly convex.
3.2.1 Duality
The OTF problem for a linear fairness notion has n + dy constraints, which is usually far fewer in number than the n² variables that make up the couplings in the primal problem. Because OTF, is also strongly convex, we will derive and then maximize the dual function instead.
Proposition 1. G has a non-emppn non-empty null-space (i.e. 3f R" + O, Gf = =Od)), then the OTF, (h) problem has a unique solution for any h : X - (0, 1]. Moreover, the OTF, problem is strongly convex and then enjoys strong duality. Its dual function is
L(X, H. Σxh, - Σ exp (
Eexp
-
Ci2 +
Eue SuGGed
(8)
with X E IR and μ Rd the dual variable vectors for the marginalization and fairness constraints.
3.2.2 Optimization
Though Eq. 8 could be maximized directly, we follow standard OT approaches 33 and perform our optimization with exact coordinate ascent. This strategy is particularly useful here, because the A, variable that maximizes L( μ1) while is fixed, is found independently of other variables in A. All variables in À can therefore be updated at the same time:
X, - elo h2 e log Eexp exp (+|-cu -c. + Encges
(9)
where we can use the stabilized log-sum-exp operation. Unfortunately, there is closed form expression for the μe that maximizes T. p1). Instead, preprocess L(A, (1) and numerically solve for each peo
Em exp (FuaGes)
Me argmin Σ (n)
(10)
with ny =
E, exp l- + x:l)-
3.2.3 Complexity
The dual problem involves two variable vectors: À with length n the number of samples and μ with length d, the number of linear fairesss constraints. It is commonly the case that dr < n, because the number of distinguished protected groups is limited. This is in contrast with the traditional OT, problem, where the dual problem involves two dual variable vectors of length
In terms of computational complexity, the update of the full has complexity O(n(n + d))). However, we stress that each A can be updated in parallel with complexity O(n + dz). When keeping the À vector fixed, we can perform a (nn)) operation to precompute the ny(A) values for the fe updates. Though these updates closed form, the inner loop of the update for μ4 only has complexity complexity Onn + dyr) for each inner loop, and should only be performed for dy variables. By permitting high tolerance the convergence of X, the necessary number of updates also limited. In our experiments, we found that single update is often enough.
We can achieve memory complexity of O(n + dy). However, we can perform efficient matrix computations by storing the cost matrix and the fairness constraints matrix G. The memory required for practical use is therefore O (n dz)).

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3.3 Adjusted OTF,
cause for concern with the smooth OTF, cost is that, as opposed to the standard OTF, does not necessarily equal zero if h is already a fair score function. Indeed, while it may be feasible to achieve (C. P)
is generally the case that the entropy H (P) 0. We would thus like find an interesting lower bound on OTF, that is tight when h F.F
Definition (OTFR,). The OTF, cost in Def. 3 is relaxed the OTFR, cost as OTFR,
min (C, - cH(P) PETT (h)
(11)
with Definition 5 (OTF.). For a score function h : X + (0, 1], the adjusted OTF, cost is computed as ( OTF, (h) OT).
II (h)
gEXn PI, h, GP' 1, / |Gh].. R4
(12)
Whereas the (h) problem involved transport fair score vectors (GP" Oay)). the OTFR, (h) problem relaxes this constraint by allowing unfairness up to the unfairness already present in h. It is thus easily seen that OTF, 2 OTFR, (h). Interestingly, because this relaxation is bounded by the unfairness of h, we have that OTFR, is a tight lower bound if h is already fair.
Proposition 2. h E F = (h) = O.
In Appendix B.I, we provide empirical intuition for this theoretical result by using OTF as a postprocessing approach and illustrate that OTF.(h) gradually converges to OTFR, (h) as the fairness of h improves.
The adjustment term OTFR, can be computed by using strong duality in similar way as OTF, (h), though now with two dual variables & E Rdd and ab E IRdr that enforce the faimess inequality, in addition to the dual variable vector, here denoted by R E IR", that enforces the row constraint on the coupling. The dual function of OTFR, is
[
L(K, Φ. v) Σ k,,, + Erelas + bb) -
_e exp
Cis t - G
with d. 0, th < 0 and Ye E, hzl For a full discussion, refer to Appendix A.4.
3.4 Minimising Unfairness
Given an efficient way to compute the parameters of the adjusted, smooth OTF2(1 cost, we can now jointly minimize the weighted sum of the classifier's loss and unfairness cost with respect to h.
Let Ly (h) denote the cross-entropy loss of h(X) for the output labels Y. To add our (TF cost this objective, we can pose the following optimization problem:
min 1 (1l-a)_yy ay (h) + aOTF(
(13)
h
with 0 S 1 hyperparameter that regulates the importance of the unfairness cost term OTF!
So far, we considered the scores vector of h as constant with respect to the OTF? (h) problem. However, the solutions for OTF, (h) and OTFR, (h) clearly depend on h. Let X", n², R², o² and s denote the fully converged dual variables. Then
θ OTF ah (T(X",²²") T(K", Φ., *b)))
(14)
ah
Even if did not update the dual variables until convergence, to approximate the true gradient [18] 16]. We further approximate the true OTF? cost by only computing it for a subset of the dataset, e.g. when batching already used to optimize (h)... The OTF (h) can optimized on a batch by only using the values of C and the columns of G that relate to these data points. For large datasets (including those we use in experiments). it is then unnecessary to precompute the C matrix between all pairs of data points, since most pairs will never actually co-occur in randomly sampled batches.
still use intermediate values

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4 Related Work
Fairness in machine learning commonly understood in two (extreme) ways 30 First, the notion of individual fairness states that similar individuals should be treated similarly [1] Second, group fairness requires that as a group, people are treated equally. The primary goal of our work is to achieve group fairness by measuring the distance of score function to set of group-fair functions. Yet, to an extent we also approach individual fairness because our unfairness quantification takes the features of each individual into account in its cost function c. By appropriately choosing it may be possible to further improve upon individual fairness in future work.
Methods that aim to improve fairness can be categorized into three types: those that perform preprocessing where the data is made more fair (25 8. as opposed those that do postprocessing where the model's predictions are modified to fit notion of fairness [21]. As a fairness regularization term, our work clearly fits in the class of improcessing methods, i.e where the machine learning model or its training is directly altered to reduce unfaimesss 7 41
40 1 [9. However, we highlight that the
OTF cost could also be used as a postprocessing approach by setting a 1.
Our work draws inspiration from two kinds of inprocessing methods. First, with those methods where (still unfair) model is explicitly projected unto the set of fair models. A popular definition of distance for this projection is the Kullback-Leile
divergence [3.23] []. Second, our work uses Optimal Transport (OT) [33 theory. Several works have applied OT to faimesss, though they almost all do so by finding barycenters [35] measures that are 'equally far' from the measures belonging to each demographic. This type of method has been applied to preprocessing [19], classification 23 42 and regression [I] 20], with the overarching goal being to move each group's 's measure closer to the barycenter. Such methods require fairness notions be encoded in the format of explicit equality, between a limited amount of groups. They do not allow for more complex constraints such as those involving continuous or multiple sensitive attributes, and make strong assumption that the cost function is distance metric.
Our proposed method instead combines the idea of projecting the model a fairness-constrained set and the use of OT to measure the cost of this projection. In this manner, our work is most similar to 34], which computes the OT projection onto such a fair set. However, they only use the method as a group fairness test. Our method has the advantage that it computes differentiable cost that can be added to the loss of any probabilistic classifier.
Experiments
Our experiments were performed four datasets from fair binary classification literature 14 with different kinds of sensitive information. First, the UCI Adult Census Income dataset with two binary sensitive attributes SEX and RACE. Second, the UCI Bank Marketing dataset, where measure fairness with respect to the original continuous AGE values and the binary, quantized version AGE_BINN BINNED. Third, the Dutch Census dataset with sensitive attribute SEX. Fourth, the Diabetes dataset with sensitive attribute GENDER. Additional information on datasets and hyperparameters is given Appendix C.2 and C.3 respectively.
28
5.1 Methods
All experiments were performed using a logistic regression classifier: To achieve fairness, we jointly optimize fairness regularizers with the cross-entropy loss as in Eq. (13) and compute the gradient of their sum with respect to the parameters of the classifier.
The OTF cost is the adjusted version from Def. 5 with e 10-3 (different choices for are illustrated in Appendix B.2). For the cost function c, we use the Euclidean distance between non- non-protected features. The fairness constraints for both sensitive features in the Adult dataset were combined by concatenating their G matrices.
As first baseline, we use the L' norm of the PDP and PEO faimess constraints and will refer to this method as the Norm cost. This simple type of fairness regularizer frequently proposed in literature 31. In our experiments, it thus represents the class of methods that can accommodate a flexible range of fairness constraints while only considering the output of a classifier and not the input features.
226 41 39
7

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Our second baseline is an OT Barycenter approach (24. This method directly minimizes the OT cost between the score distributions of two sensitive groups. However, the implementation can only minimize PDP, and does not admit composition of linear fairness notions or continuous attributes. We therefore only minimize unfairness with respect to SEX and AGE_ BINNED. To show the drawbacks of this limitation, we also report PDP unfairness with respect to RACE and the continuous AGE.
The third baseline Unfair, which was a classifier trained without fairesss cost term.
5.2 Evaluation
To measure the predictive power of the methods, the ROC AUC score was measured. Violation of PDP is quantified as the absolute value of the maximal Pearson correlation between the scores h(X) and each dimension of the sensitive attributes S.- For PEO, the violation is computed as the maximal PDP for the data points of each one-hot encoded output label Y,. These measures mirror the requirements in Eq. (T) and Eq. (2) that the covariances ought to equal zero, though we choose the Pearson correlation instead such that the violation is normalized between 0 and 1. Thus we avoid that a method may improve on linear fairness notions by squeezing the variance of its scores.
All methods were tested for various strengths of a (see Eq. (13) Each configuration of each method (i.e. each a value and fairness notion) was tested for 10 train-test splits with proportions 80/20. We report mean test set scores in Fig. I and show the confidence ellipsd" of their estimator for the first standard deviation. The train set results are reported Appendix B.3 and show the same trends.
Note that for the OTF and Norm regularizers, we omit fairness scores for notions that were not optimized in that configuration. While SEX and RACE minimized at the same time for the Adult dataset, the AGE and AGE BBNNEDNNNDDDDDDD attributes were minimized in separate training runs.
All experiments were conducted using half the hyperthreads on an internal machine equipped with a 12 Core Intel(() Xeon(R) Gold processor and 256 GB of RAM. In total, all experiments, including preliminary and failed experiments, used this machine for approximately 200 hours.
5.3 Results
On all datasets, our OTF method makes similar trade-off between AUC and PDP fairness as the baselines for various a values. However, OTF clearly outperforms Norm for both sensitive attributes on the Adult dataset in Fig. Ia when they are trained to minimize PEO, especially in the lower violation range. Similarly, OTF achieves better trade-off between AUC and PEO violation on the Diabetes dataset in Fig. Ld. especially on the train set (see Appendix B.3). This advantage in minimizing PEO may be due to the fact that OTF has the incentive to achieve fairness by exchanging classifier scores between similar individuals. Conditioning on the output label rewards such exchanges, individuals with similar labels can be expected to have similar features.
Furthermore, we remark that our OTF method is as flexible as the Norm baseline in minimizing either PDP or PEO with respect to multiple sensitive attributes on the Adult dataset and the continuous AGE attribute on the Bank dataset in Fig [b Implementations of differentiable regularizers that can accommodate such attributes are scarcely available. For example, the Barycenter method was not trained to improve fairness with respect to RACE, its implementation can not handle multiple sensitive attributes. Since it can only minimize the OT cost between a limited number of score distributions, it is naturally ill-equipped for continuous attributes. In separate runs the Bank dataset, we therefore also trained OTF and Norm to minimize unfairness with respect to THE_BIINED.... Though the results are subject to noise, we observe similar trends for the other non-continuous attributes: our method is clearly on par with the baselines. On the train set results (see Appendix B.3) the OTF method significantly outperforms the Norm baseline for PEO.
On the Adult, Bank and Diabetes datasets, the proposed OTF method thus displays its advantage in both an effective AUC-fairness trade-off, and flexible capacity for fairness. However, we note that on the Dutch dataset in Fig. Ic no method seems to display a significantly better trade-off. We hypothesize that no such advantage is seen here because PEO, the fairness notion where our method usually achieves superior results, may be relatively easy to satisfy on this dataset.
Retriieved from https: /gi thub. http:/depmmidd//asseeeeeee htttierrsttiinaaaeeeeeeeeeeeeeee fairness (Apache License 2.0) We use the variance of the estimator of the mean, which is smaller than that of individual scores
8

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OTF
Norm
Barycenter
Unfair
0.90
ROC AUC
0.88
0.86
64
0.84
0.00
0.10
0.20
0.30
0.10 Violation of PDP race
0.05
0.10
0.20
0.03 0.05 0.08
0.10
Violation of PDP -ex
Violation of PEO sex
Violation of PEO race
(a) Adult dataset
0.93
ROC AUC
0.92
0.08 Violation of PDP age
0.03
0.05
0.01
0.02
0.03
0.02
0.04
0.06
0.03
0.04
0.05 0.06
Violation of PDP age binned
Violation of PED age
Violation of PED age sinnnin
(b) Bank dataset
0.88
0.61
ROC AVC
ROC AUC
0.86
0.60
0.84
0.59
0.00
0.10 0.15 Violation PDP-sex
0.05
0.02
0.04 0.06
0.08
0.01
0.02
0.03
0.04
0.02
0.04
0.06
Violation PEO
Violation PDP gender
Violation of PEO gender
(c) Dutch datasct
(d) Diabetes dataset
Test set results for the methods that were trained reduce the evaluated fairness measure (PDP or PEO). Violation of PDP (and PEO) is computed as the maximal absolute Pearson correlation between the probability scores (conditioned on the output labels) and each sensitive attribute.
Figure I:
5.4 Limitations and Impact
We nuance our results by pointing out that the OTF method has higher computation complexity (O(n(n + dy))) than Norm, which has complexity O(ndz) with the batch size and dy the number of fairness constraints. Furthermore, as discussed Sec. 2.1 we only evaluate linear fairness notions and leave extension non-linear notions of fairness [ for future work.
The fairness of a system should always be judged based on holistic consideration of the context of the system (3) and relevant ethical principles 4 Without it, any method to improve faimess may reinforce the underlying injustice that led to the risks of discrimination that we aim to solve 22
6 Conclusion
In this paper, we constructed the Optimal Transport to Fairness (OTF) method as differentiable fairness regularization term, which combines the advantages of flexible linear faimess constraints with Optimal Transport theory, thereby taking non- non-protecn protected similarities between individuals into account. In experiments, we show that our final method achieves similar AUC-fairness trade-off as other cost terms for notions of fairness inspired by demographic parity, and significantly better for notions inspired by equalized odds. At the same time, OTF empirically displays why its increased flexibility, i.e. its ability improve all linear fairness notions with respect to multiple sensitive attributes that may be categorical or continuous, provides an advantage over previous faireess applications of OT.

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In the future, we hope to further investigate the properties of OT-based fairness, inspired by its clear advantage in achieving equalized odds. Though our method was only applied linear fairness notions, OT may provide opportunity to better achieve non-linear notions of fairness, e.g. through a creative choice of the cost function that directs the transport of classifier scores.
Acknowledgments and Disclosure of Funding
The research leading to these results has received funding from the European Research Council under the European Union's ' Seventh Framework Programme (FP7/2007-2011 (ERC Grant Agreement no. 615517), and under the European Union's s Horizon 2020 research and innovation programme (ERC Grant Agreement no. 963924), from the Flemish Government under the "Onderzoeksprogrammn Artificiële Intelligentie (AI) Vlaanderen" programme, and from the FWO (project no. GOF9816N, 3G042220). MB is supported by doctoral scholarship from the Special Research Fund (BOF) of Ghent University (reference number: BOF20/DOC/144)
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] In Sec. we proposed the OTF cost term and deployed tricks to minimize it effectively. In Sec. 5 we show that the results of using our proposed method are close to the baselines or surpassing it.
(b) Did you describe the limitations of your work? [Yes] We describe limitations of our considered fairness notions in See. 2. and the complexity of our proposed OTF cost term in Sec. 3.2.3 We briefly discuss overall limitations of faimess methods in Sec. 5.4
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(c) Did you discuss any potential negative societal impacts of your work? [Yes] We briefly discuss the danger that inconsiderately applied fairness methods can pose in Sec, |.4
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Our method is constructed in several steps. When required, discuss and add additional assumptions.
(b) Did you include complete proofs of theoretical results? [Yes] Proofs of all proposi- tions and derivations of other results are included in the Appendix.
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or URL)? [Yes] There references to the datasets we used in the Appendix. Code and a pipeline to run it are included in the supplemental material.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Initial training details are given in 5.2 More details of the hyperparameters are included the Appendix.
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] Since show the trade-off between two goals (accuracy and fairness) we show the confidence ellipses.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] This is mentioned at the end of Sec. [.2
4. If you are using existing assets (c.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] For the Barycenter code, a reference to the Github implementation is included. For datasets, we refer to the UCI repository.
(b) Did you mention the license of the assets? [Yes] For the Barycenter code, we include the license. We refer to the data according to the requested citation policy.
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [No] The datasets we use are popular, public datasets in machine learning fairness literature. Obtaining consent is infeasible. However, the data is highly anonymized and its popularity promotes reproducibility in the community.
(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [Yes] As briefly mentioned in the Appendix, the data is anonymous and free from any personal identifiers.
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "z8XIgSlBXws", "reviewer": "Reviewer_Sans", "summary": "This paper proposes to formulate the unfairness as the smallest OT cost between the score function of the classifier and the closest fair score function over the same data. The author proposes to compute the OT cost as differentiable fairness regularizer, and validates the effectiveness of their method on two datasets.", "questions": "In line 267-268, 'For the cost function c, we use the Euclidean distance between non-protected features', does it mean that the optimal transport is between training distribution and some weighted training distribution, i.e. during training, the training distribution is adjusted, instead of predictions of training samples?", "limitations": "The limitations and potential negative societal impact are adequately addressed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nThe overall formulation of unfairness and OT cost is well-established. The idea of formulating unfairness as smallest OT cost is novel, and the relaxation of OT regularization is also sound. The author also analyzes the computational complexty and compare it with baseline method of L1 linear constraints.\n\nWeaskness:\n\nMy main concern is the high computational complexty. On large datasets, applying the proposed method seems to have much longer training time than Norm baseline. Also, computing the OT regularization requires more computational resources than Norm baseline. Can the author present a case on the running time of the proposed algorithm on large-scale applications?\n\nDetails of the proposed method are not clear. For example, it is not clear how the cost matrix in Eq. 7 is calculated; it is also not clear which part (training loss, training distribution, or predicted score) is adjusted during training to achieve fairness.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "XE48sMzo669", "reviewer": "Reviewer_cFmV", "summary": "Traditionally, fairness in machine learning force the prediction of classifiers to have similar statistical properties for different demographic groups. An example is rescaling the score function of the classifier. However, such rescaling ignores the similarities and dissimilarities between members of different groups. To this end, the authors introduce OTF (Optimal Transport to Fairness) to quantify the unfairness of a probabilistic classifiers, which is defined as the smallest optimal transport cost between the score function of the classifier to a fair score function over the same data. In addition, OTF can be computed as a differentiable fairness regularizer for a class of linear fairness definitions. In experiments, OTF shows better fairness-prediction trade-off.", "questions": "- For different OT methods, it is unclear whether for a fixed ROC AUC score, violation of fairness metric is different. Or is it the case that both ROC AUC score and fairness violation score are different for different OT methods?\n\n", "limitations": "Limitations are addressed in the paper.", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength: \n\n- The paper is well structured. The motivation is clear.\n\nWeakness: \n\n- A motivating example showing the computation of OTF for a classifier will highly improve the readability of the paper. In the final edits, such an example should be added.\n- Experiments are limited as the authors consider only two datasets.\n\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "IQt0L-IchJT", "reviewer": "Reviewer_MHSV", "summary": "This paper proposed to quantify the unfairness of a classifier in machine learning by the Optimal Transport (OT) theory. By introducing the definition of *linear fairness notion* under the probabilistic perspective, the author proposed to quantify the unfairness with OT cost between a classifier and a set of linear fair score functions. With the above definition of linear fairness notion, the fairness regularizations are reformulated as the constraints of this OT problem. Based on this theoretical result, the author applied entropic smoothing and debiased methods to solve the above OT problem. The empirical results on two datasets (i.e., *Adult* and *Bank*) indicate that the proposed methods achieve a better AUC-fairness trade-off under the demographic disparity (DP) and equalized odds (EO) metrics, compared to several selected baselines.", "questions": "1. In Line 83, it is unclear why *\"multiple linear fairness notions can be combined into a new linear fairness notion by stacking their respective g functions\"*. The expression *\"stacking\"* is not mathematically clear. What does it mean? The element-wise multiplication of g functions or addition? Could you explain more about this claim?\n2. The authors only provide the train set scores in the main body of this paper and leave the test set results in the appendix. The author claimed that test results are with large noise due to the test set size. But I still think it is crucial to provide all test results in the main body of the paper. Although these results may be noisy and \"not so beautiful\", but the authors still need to analyze these results and provide discussions (e.g., w.r.t. the stability and the variance of these results).\n3. About the adjusted $OTF_{\\epsilon}$. A common solution to debias the entropy-regularized OT problem is the unbiased Sinkhorn divergence [2], which is  defined as $SOT_{\\epsilon}(\\mu, \\nu) \\triangleq OT_{\\epsilon}(\\mu,\\nu) - \\frac{1}{2}( OT_{\\epsilon}(\\mu,\\mu) +  OT_{\\epsilon}(\\nu,\\nu))$. Could the author discuss the difference between the proposed $OTF_{\\epsilon}$ and the debiased Sinkhorn divergence, and emphasize why the adjusted $OTF_{\\epsilon}$ is preferred in this work?\n4. How about the experimental results on other datasets (e.g., Toxicity Classification [3])? It will be better to provide more empirical results on different datasets to support the consistently better performance of the proposed method.\n5. About the applicability on high dimensional data. In this paper, only logistic regression model is used as the probabilistic classifier. The reviewer understands that it is because the feature dimension in Adult and Bank datasets is very low ($d_{\\mathcal{X}}\\leq 20$ if I remember correctly). In the scenario of high dimensional data (e.g, images data like CelebA [4], text data like [3]), the one-time computation of the transport cost matrix $C$ (e.g., like the Euclidean distance used in this paper) on original space seems no longer feasible. However, if we compute the cost matrix with the embedded representation from a deep neural network, will the proposed method still be applicable? Will it introduce more computational overhead? Or what kind of modifications do we need?\n\nminors:\n- Could the authors discuss the feasibility or limitation of their proposed method for the *sufficiency rule* (which encourages $Y\\bot A|\\hat{Y}$)?\n- It would be better if the author could carefully check the typos for the future version. For example in Eq.(2), it seems it should be \"$\\forall l \\in \\{0,1\\}$\", but the notation \"$\\in$\" was missed; missing space between words in Line 100.\n\nReferences:\n- [1] Chiappa, Silvia, et al. \"A General Approach to Fairness with Optimal Transport.\" AAAI. 2020.\n- [2] Genevay, Aude, Gabriel Peyré, and Marco Cuturi. \"Learning generative models with sinkhorn divergences.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2018.\n- [3] Jigsaw. Jigsaw unintended bias in toxicity classification, 2018.\n- [4] Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Large-scale celebfaces attributes (celeba)\ndataset.", "limitations": "- The main body of this paper only shows the results on the train set. More analysis and discussion on test results are needed.\n- The feasibility or limitation for the regression problem is not discussed in this paper.\n- The applicability to real-world high-dimensional data (e.g., images, text) has not been discussed in the current version.\nSee the Questions and Weakness part for details.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. This paper proposed new concepts, i.e., *Linear Fariness Notion* and *Optimal Transport to Fairness (OTF)*, which provide a new perspective to introduce the fair constraints during the learning process. The derivation seems correct.\n2. Compared to the previous OT-based related work, this paper proposed to implement the OT-based fairness regularizer for DP and EO with a differentiable manner. And this paper is also applicable for the scenario of multiple sensitive attributes.\n\nWeakness:\n1. The applicability to the regression problem was not discussed in this paper, while other OT-based related work can also be applied to regression problems [1].\n2. The author only conducted the experiments on two datasets, which seems not enough. More empirical results on other datasets (e.g., Toxicity Classification [3]) are expected.\n3. The paper only provides the experimental results on train sets in the main body, and provides limited analysis of the test results in the appendix.\n4. The applicability to high-dimensional real-world data is not discussed.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept