fairret: a Framework for Differentiable Fairness Regularization Terms
Authors: Maarten Buyl
MaryBeth Defrance
Tijl De Bie
Venue: ICLR 2024 poster (2024)
Keywords: fairness
statistics
differentiation
regularization
classification
Primary area: societal considerations including fairness, safety, privacy
TL;DR: We represent fairness notions as an equality between statistics with a general (linear-fractional) definition. We propose differentiable regularization terms to then pursue these fairness notions in a modular, simple pipeline.

Abstract: Current tools for machine learning fairness only admit a limited range of fairness definitions and have seen little integration with automatic differentiation libraries, despite the central role these libraries play in modern machine learning pipelines.

We introduce a framework of fairness regularization terms (fairret) which quantify bias as modular objectives that are easily integrated in automatic differentiation pipelines. By employing a general definition of fairness in terms of linear-fractional statistics, a wide class of fairrets can be computed efficiently. Experiments show the behavior of their gradients and their utility in enforcing fairness with minimal loss of predictive power compared to baselines. Our contribution includes a PyTorch implementation of the fairret framework.

Published as a conference paper at ICLR 2024
FAIRRET: A FRAMEWORK FOR DIFFERENTIABLE FAIR- NESS REGULARIZATION TERMS
Maarten Buyl
MaryBeth Defrance
Tijl De Bie Ghent University
Ghent University maarten.buyl @ bgentt.e
Ghent University marybeth.deffrnnee defrannce @ ugent.be
tijl.debi@uugent
ABSTRACT
Current fairness toolkits in machine learning only admit limited range of faimess definitions and have seen little integration with automatic differentiation libraries, despite the central role these libraries play in modern machine learning pipelines. We introduce a framework of fairness regularization terms (FAIRRETS) which quantify bias as modular, flexible objectives that are easily integrated in auto- matic differentiation pipelines. By employing a general definition of fairness in terms of linear-fractional statistics, a wide class of FAIRRETS can be computed efficiently. Experiments show the behavior of their gradients and their utility in enforcing fairness with minimal loss of predictive power compared to baselines. Our contribution includes PyTorch implementation of the FAIRRET framework.
INTRODUCTION
Many machine learning fairness methods aim enforce mathematical formalizations of non- discrimination principles (Mehrabi et al.] 202T), often by requiring statistics to be equal between groups (Agarwal et al., 2018). For example, may require that men and women receive positive decisions at equal rates in binary classification (Dwork et al.] (012). The main interest in fairness tools is to meet such constraints without destroying the accuracy of the MI model.
large class of these fairness tools utilizes regularization terms, i.e. quantifications of unfairness that be added the existing error term of an unfair ML model (Kamishima et al., 2012 Berk
ett al. 2017 Zafar et al. [2199] Padala & Gujar [021] Padh et al.] | [uyl & De Bie 2022 The
modularity of such loss terms appears to align well with the paradigm of automatic differentiation
libraries like PyTorch (Paszke et al.) 2019), which have become the bedrock of modern machine
learning pipelines. However, the practical use this modularity has seen little interest thus far.
Contributions
Hence, we formalize a framework of fairness regularization terms (FAIRRETSS) that consolidates research in differentiable fairness methods. FAIRRET quantifies a model's unfairness as single value that is minimized like any other objective through automatic differentiation.
We implement two types of FAIRRETS: FAIRRETS that directly penalize the violation of fairness con- straints and FAIRRETS that minimize the distance between model and its projection onto the set of fair models. These FAIRRETS support any fairness notion defined through mnear-fractionl statisties (Celis et al. 2019), which is far wider range than the exclusively linear statistics typically consid- ered in literature (Zafar et al. [019 Agarwal et al., 2018) 8). Moreover, our framework generalizes to the simultaneous handling of multiple sensitive traits and (a weaker form of) fairness with respect to continuous sensitive variables. By design, FAIRRETS are both modular and extensible such that future work that can benefit from their wide applicability. Appendix OppendixEEconnainn
code example.
We visualize the FAIRRETS' gradients and evaluate their empirical performance in enforcing fairness notions compared to baselines. We infer this is far more difficult for faimess notions with linear- fractional statistics, which were rarely studied in prior work, than those with linear statistics.
The framework is available package at htttppss//ggtthbb..oo//../e///
fairrett
Related Work Fairness tools are classified as preprocessing, inprocessing or postprocessing MMehrabi al. 2021). FATRRETS perform inprocessing, they are minimized during training.

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A popular approach to faimess regularization is to penalize the violation of fairness constraints (Zemel et al. 2013 Padala Gujar 2021 Wick et al. 2019), which we formalize as a FAIRRET. We
also take inspiration from postprocessing methods that project classifiers onto a fair set (Alghamdi
et al. 2020 Wei et al. 2020) and penalize the cost of this projection (Buyl & De Bie 2021) as a
FAIRRET. Fair representation learning (McNamara al. 2019 Oneto al.) 2020 Franco al.,
2022) finds intermediate representations that minimally contain sensitive information. An example
the adversarial approach of Adel al. (2019), which is a baseline in our experiments.
Celis et al. (019)) observed that many fairness definitions express parity between linear-fractional statistics. They propose a meta-algorithm to find optimal classifiers that satisfy this constraint. Instead, we employ simpler (yet sufficiently expressive) Innear-fractions form and propose an algorithm to use them in the construction of linear constraints that does not require meta-algorithm.
Popular faimess toolkits such as Fairlearn (Bird et al., 2020) and AIF360 (Bellamy et al. 2018) expect the underlying model in the form of scikit-learn Estimators that can be retrained at-will in fairness meta-algorithms. Instead, our proposed FAIRRETs act a loss term that can simply be added within a training step. The aforementioned toolkits have some integration with automatic differentiation libraries in adversarial fairness approaches (Zhang et al., 2018), yet these still require full control over the training process and lack generality in the fairness notions they can enforce.
Two PyTorch-ppecific projects with similar goals as our paper are FairTorch (Masashi 2020) and the Fair Fairness Benchmark (FFB) (Han et al.] (023). However, neither present a formal framework and both only support a limited range of fairness definitions.
2 FAIRNESS IN BINARY CLASSIFICATION
In fair binary classification, are provided with random variables (X,S,Y) with X Rd the feature vector of an individual, e Rd, their sensitive feature vector and E {0. 1} the binary output label. In what remains, all expectations are taken over the joint distribution of (X,S,Y).
The goal is to learn a classifier f such that its predictions f(X) match while avoiding discrimi- nation with respect to S. In this section, we will assume directly provides binary decisions, i.c.
f: Rd. - {0, 1}, as this is expected in traditional formalizations of fairness. However, since such 'hard' classifiers not differentiable, we will instead be learning probabilistic classifiers in Sec. 3
Further note that our definition of sensitive features S as real-valued and ddddimensional vectors generalization of typical fairness definitions which assume a categorical (or binary) domain for sensitive features (Verma & Rubin 2018). We will one-hot encode such categorical traits, e.g. by encoding 'white' or 'non-white" as the vectors S = (1, and S (0. 1) respectively. Our generalization allows us to take multiple non-exclusive sensitive traits into account by mapping them to different values Sş in the same vector S for E [d,] = (O.... d, - 1}. Additionally, by letting S& R, we allow soft specifications of identity rather than requiring hard discretization.
2.1 PARTITION FAIRNESS
Though we will allow any feature vector S Rd. in our framework, popular fairness definitions require every person to belong to exactly one demographic group. We call this partition fairness.
Definition 1. In partition fairness, S is a one-hot encoding, i.e. Sy {0, 1) and Eke[d.] Sk. 1. Example A straightforward, popular definition in partition fairness is Demographic Parity (DP), also known as statistical parity (Dwork el al. 2012 Verma & Rubin 2018). It enforces
Vk [d,] P(f(X) = = 1))== 1)
which states that all groups ought to get positive predictions at the same rate (i.e. the overall rate).
Let yk:
B[Sxf(X)] is easily shown that y(k; PX) = S₃ = 1). Thus also
3|1a1
P((()) = = 1) =) = = (k: f) B[)()).
(2)
httpss//sciiitt-eerm..co
htt.m//wrggl..///aeeee/////////p describes these Estimators
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Table I: Fairness definitions and their and & functions. Conditional Demographic Parity encom- passes many notions with an arbitrary function & conditioned on the input X.
Faimnsss Definition
Bo
01
B1
Demographic Parity (Dwork et al.
2012)
0
0
Conditional Demographic Partt
(Wachter et al.
2020)
0
5(X)
5(X)
0
Equal Opportunity (Hardt al.
2016
0
Y
Y
0
False Positive Parity (Hardi et al
20T6
0
I-Y
1-Y
0
Predictive Parity (Chouldechova
2017)
0
False Omission Panty
Y
-Y
1
-1
Accuracy Equality (Berk et al.
2021
- Y
2Y
I
Treatment Equality (Berk et al.
2021
Y
-Y
0
1-Y
In Example [ fairness formalized by requiring statistic y to be equal across groups. This prin- ciple can be generalized to a wide class of parity-based fairness notions. In particular, we consider those expressed through fnear-fractional statistics (Celis et al.) (209)).
Definition 2. A finear-fractional statistic computes values yk: f) E for sensitive variable Sk and classifier f Rdd {0, 1}. We assume y differentiable with respect to f. takes the form
(X
f(: B[S ((X)
with ag, a1 Bo. and B1 all functions that do not depend on or f. Let I denote all such statistics. (XX) Also, let f(f) (XX
denote the overall statistic value without conditioning on S. Definition 3. A fairness notion is expressed through statistic T.T. The set Fy of classifiers that adhere the fairness notion is defined
Fy {f Rdd - {0, VK [d,]
(4)
i.e. the statistic y(k; f) for each Sy. equals the overall statistic 7().
Indeed, the DP fairness notion Example D is expressed as a fairness notion as defined in Def.
with ihear-fractional statistics as defined in Def. g The same holds for the following notions.
Example Equalized Opportunity (EO) (Hardi el al. 2016 only computes DP for actual positives
== 1. Its statistic is thus the recall P(f(X) = l 1,5x = 1), i.e. y(k; = 3[S,F]
((S))
Example 3. Predictive Parity (PP) [Chouldechoval 2017), which compares the precision statistic
P(Y f(X) 1, 1), i.e. f(: E1S
[[Sxf(X)]
Example 4. Treatment Equality (TE) (Berk et al. 2027) balances the ratios of false negatives over
false positives, i.e. (k: f) E[S.
[[Sa (XJJ Unlike the other notions, its y is not a probability.
Table I summarizes the a and B functions several fairness notions (Verma Rubin 2018 with fractiona statistics. Their derivations found in Appendix A.
Definition 4. A linear-fractional statistic E I is linear when B1 (X,Y) =
Ler L1 c denote the set of all linear statistics.
Fairness notions with linear statistics y E IL are thus identified in Table [ by checking the column for B1. Such notions especially useful because the faimess constraint in Def. B] easily written as a linear constraint over classifier f. In turn, this makes the set of fair classifiers Fy a convex set, which leads convex optimization problems (Boyd Vandenberghe, 2004). Thus, the constrained optimization of f can be efficiently performed if f is itself linear (Zafar et al. 2019).
However, faimess notions with linear-fractional statisties y (r. do directly lead linear constraints in Def. 3 To facilitate optimization, we therefore propose to narrow the set of fair classifiers T, the subset where the statistics are all equal in a particular value c.
fairness notion is expressed through linear-fractional statistic y E r such that the set (c(c) of classifiers f that adhere the fairness notion defined as
Definition 5. Fix a E R. A fairnes
Fc( & RS:Rd. (0. Vk [d,] yk: =c}
(5)

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Proposition 1. With y r, the e-fixe notion Ty(c) enforces linear constraints:
y(: =c = [[Sx(a((,) Y, c) (X))(XXXXXXXX.. where a (X,,Y,c = (x, ca,, (X. and B(X. Y, Bo(X, cB)(X.
(6)
Using Prop. 1 we can still obtain linear constraints for faimess notions Fy with Anear-fractional statistics E E IL by considering their c- c-fixed variant Fy(c) instead. This sacrifices a degree of freedom because statistics y(k;)) are no longer allowed to be equal for any overall statistic F(), they must now do so for the specific case where 7() = c. However, there are values that still lead interesting sets (c). In the FAIREETS we propose, we take an unfair classifier and fix 7(h) to construct the set of all fair classifiers (T))) that would result from fair redistribution of scores in h over the sensitive groups
Though Prop. iis inspired by Celis et al. (2019), our use of this result vastly differs. Instead of fixing the statistics to a single value c, they set many pairs of upper and lower bounds for each group's s statistics, giving rise as many optimization programs. They then propose meta-algorithm that searches the best classifier over each of these programs. A meta-algorithm is not necessary in our framework, we will allow e evolve during training. While we have no formal convergence guarantees for this approach, empirical results show it works well in practice.
2.2 BEYOND PARTITION FAIRNESS
Having firmly rooted our definitions in partition faimess (Def. D. we now abandon its assumptions. First, we allow Sx E R. Second, we extend to multiple sensitive features with Σ Sy E R.
2.2.1
CONTINUOUS SENSITIVE VALUES
Admitting continuous values for someone's sensitive trait, i.c. Sk E R allows us to take naturally continuous features, such as age, into account. Also, it provides an opportunity for an imprecise specification of demographic group membership.
For instance, instead of exactly knowing the gender of an individual, may only have a probability available, e.g. because it is noisily predicted by a third-party classifier, or to protect the individual's privacy. By allowing Sk (0, 1). the attribute Sp could then express 'woman-ness' instead of a binary 'woman' or "not woman'. Thus, we also allow individuals to themselves quantify how strongly they identify with group, rather than requiring binary membership.
Our notation already generalizes to non-binary Sx values; they can simply be filled in for linear- fractional statistics y E as defined in Def. Fairness as formalized in Def. can then still be enforced through y(k; y(f).
Remark 1. Partition fairness constraints stem from the principle of treating distinct groups equally.
This does not directly apply for non-binary Sk- For example, if there is only one, continuous sensitive variable (So) such as the age of an individual, then we cannot compare y(0: f) to another group's 's statistics. Instead, y(0; f) must be compared to value independent of S.
Enforcing ykk, ff = -f) then sensible choice, as satisfies key properties one can expect from fairness measure. First, the constraint is met when St = s, i.e. when Sp. is deterministic constant. Second, holds if S& has no linear influence on the numerator and denominator of y, i.e.
cov(Se. (X, (X))ooXX, Y)) = cov(Sz. an (X, Y) + ((X)i Y)) = =
For a full derivation of this result, we refer Appendix B.1
2.2.2 MULTIPLE AXES OF DISCRIMINATION
By allowing Σs Sg E R. support that S contains information about people from several sensitive traits, e.g. gender, ethnicity, and religion. Because these each form a possible axis of discrimination, can 'sum' these sources of discrimination by combining the constraints.
For example, if pairs of sensitive features (So. S1) and (S2, S3) each partition the dataset, then fair- ness requires both y0: f(: 7(f) and y(: y(3:f) y(f). Combined, these constraints make up the fairness definition in Def. 3 The use of one-hot notations for sensitive values thus already allows to combine axes of discriminationn for categorical sensitive traits.
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Remark 2. An important limitation is that only view faimess separately per axis of discrimina- tion. Outside the partition fairness setting, this means that some intersections of sensitive groups, e.g. 'black woman'. will not be represented in the constraints that enforce fairness with respect to 'black' and 'woman' separately (Kearns et al. 2018 A toy example is given in Appendix B.2
3 FAIRNESS REGULARIZATION TERMS
The popular approach to modern machine learning is to construct pipelines consisting of modular, parameterized components that are differentiable from the objective to the input. We therefore use probabilistic classifier models h R2 - (0. 1) from now on, where decisions are sampled from a Bernoulli distribution with parameter h h(X). Let H denote the hypothesis class of these models.
Remark 3. Fairness statistics y(k; h) over the output of probabilistic classifier h only approxi- mately verify their respective fairness notions, as these were only defined for hard classifiers with binary output (Lohaus et al. (2020). In Appendix B.3] we discuss the impact of this approximation and how its fidelity can be traded-off with the quality of the gradient of y(; h) with respect to
In binary classification, we minimize loss Ly (h) over the probabilistic classifier h given output labels Y, e.g. the cross-entropy. In fair binary classification we additionally pursue E Ty:
min Ly heT,
For finear-fractions statistics, the constraint is linear when considering the c-fixed variant of Fy (using Prop. ID. However, for con-couvex models h, the constrained optimization of h will remain non-convex as well. In the general case, we thus relax h E Ty and instead incur a cost to h $ Ty. Definition 6. A fairness regularization term (FAIRRET) (h) H - Rzo quantifies the unfair- ness of the model h E H with respect the fairness notion defined through statistic y.
A FAIRRET is strict it holds that h E TTy
(h) =
The objective in Eq. 7 is then relaxed
min (h) + (h)
(8)
ht
with d a hyperparameter. The objective in Eq, ( is equivalent to Eq. 7 Io is strict.
Remark 4. We call Ry regularization term, yet its purpose is not reduce model complexity or improve generalization performance, in contrast to traditional regularization in machine learning Kukačka al. 2017). Instead, we aim to limit the hypothesis class of h to the set of fair classifiers.
In what follows, we introduce two archetypes of FAIRRETS: violation and projection. We visualize R, for each FAIRRET in Fig. with y the positive rate statistic (thereby enforcing DP).
3.1 VIOLATION FAIRRETS
To quantify ht E Fy, start from the violation v(h) of the constraint that defines Fy:
y(k; h) 7(h)
(h)
(9)
with V H - Rdd a valued function with components Vv. Clearly, v(h) 0 - h Fy,
Note that v(h) is normalized? downscaling its statistics y without reducing relative differences between groups (Celis et al. 2019).
2 by 7(h) such that classifier cannot minimize v(h) by uniformly
Definition 7. We define the Norm FAIRRET as (h) = ||v(H)]], with ||-I| norm over Rd.
Many variants of the Norm FAIRRET have been proposed, e.g. by Zemel et al. (013)) Padala & Gujar (02T)) Wick et al. 2019) and Chuang & Mrouch| (2020). However, fairness evaluation metrics often only consider the maximal violation. Hence, we propose the SmoothMax variant.
In cases where 7(h) = O, can simply use (h) (y; k)] instead. We assume h(X) E (0, 1), so this only occurs in degenerate cases for the notions in Table T (like when all Y 0 for Equal Opportunity).
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Max
Dra-projection
Dis-projection
Deen-projection
individuals with So
individuals with S
7(A)
711:)
Ra
0.0
-0.5
7(A)
(0;
-1.0
0.0
1.0
0.6
0.8
1.0
Figure I: The model was trained on the ACSIncome dataset without FAIRRET (i.e. A 0) and ends up with disparate positive rates y(:: h) 7h) > y(1: h) for the one-hot encoded sensitive variables (So. Si). These should be brought closer to the overall positive rate f(f). We show probability scores h and the gradients of several FAIRRETS R, with respect to The gradients are normalized by dividing them by their maximum absolute value per FAIRRET and per group. They are positive for samples with So 1, implying their scores should decrease, and vice versa for Sj 1.
Definition 8. We define the SmoothMax FAIRRET as (h) - log Eke[d.] (xeedd.jepp(vvc (h)) log d,
Because the SmoothMax performs the log-sum-exp operation over the violation, it can be considered smooth approximation of the maximum. We subtract log d, ensure the FAIRRET is strict.
Generally, violation FAIRRETS can be characterized as functions of the violation v(h). This lends them interpretability, but it also means that the gradient Ry decomposes as
dv = ah
V
V.R,
(10)
with 26 the Jacobian of v(h). The gradients of violation FAIRRETS Ry thus only differ in R, gradient. Hence, the Norm FAIRRET is excluded from Fig. [eecause its gradients equal those of SmoothMax after normalization. Figure also suggests that violation FAIRRETS convey little information how each individual h(X) score should be modified. Instead, they merely direct scores to uniformly increase or decrease within each group.
3.2 PROJECTION FAIRRETS
Recent postprocessing approaches to fairness redistribute all individual probability scores of a model h( X) to a fair scores vector with a minimal loss in predictive power. For example, Alghamddi et al. (2020) project the scores onto the fair set Fy as a postprocessing step. Yet. the cost of this projection can be seen as quantification of unfairness that may be minimized as FAIRRET during training.
Given statistical divergence or distance we can generally define such projection FAIRRET as (11)
(h) min B[D(F(X) ((X)). feT,((())
Importantly. we do not project h onto the general fair set Ty. but the c-fixed subset Ty(c) with = (h)) The c-fixing is done such that the projection only requires linear constraints for linear- fractional statistics (see Prop. D projection FAIRRET (Eq. TTp is then convex optimization problem if we limit ourselves to D that is convex with respect to f. which is the case for all projections discussed here. In particular, we c-fix the overall statistic 7(h) of h because this ensures h can always be projected onto itself if is already fair, as then h E F, (hh)).
There is some abuse of notation here. When taking the gradient Jacohian with respect to h, take it with respect the vector of outputs of h for of input features sampled from the distribution over X.

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Definition 9. The DEL Pxo-projettion uses the binary Kullback- divergence
f(X) hX)
(1 (()) log 1 f-)
(fx(((X) |I h(X)) = f(X) log
(12)
1- h(X)
The DKL -dierrennce is both Csiszar divergence and Bregman divergence (Amari 2009). Also,
the cross-entropy error minimized in Ly (h) equals DKL (Y || h(X)) up to constant. The minimiza-
tion of Eq. (8) thus comes down simultaneously minimizing DKL between h(X) and the data Y, and between hX) and the closest f Fy (7))) (Buyl & De Bie 2021).
Definition 10. The Dis-projection uses the binary -hasennn divergence.
(is((()) II h(X)) 2 =xx m(X) DKI (h(X) m())
1
(13)
with m
m(X) (X) (X) ((X)..
Just like DXL, the Dis-divergence is a Csiszar divergence. However, the Djs-divergence is symmet- ric with respect to its arguments f and h. which is not the case for the diveereeeee
Definition 11. The Dsep-prajection uses the squared Euclidean distance between the two points (1 - ((X) and (1 - h(X), h(X)):
DSED ((X) h(X)) & 2((() - -(X))²
(14)
DSED is a Bregman divergence between the Bernoulli distributions with parameters f(X) and h(X).
In practice, we evaluate projection FAIRRETS (h) in two steps.
f = arg min B[D(((X) II A(X))]
(15)
(7(h)
iii R = B(D(f (X) II h(X))
(16)
While keeping h fixed, step (i) computes the overall statistic 7(h) and then finds the projec- tion f through constrained optimization. Subsequently, step (ii) keeps f* fixed and computes (X)( || h(X)] as function of h, which we use to compute the gradient with respect to h. This gradient differs from the actual gradient of the optimization as function of h in projection FAIRRET (Eq. 1T). because the latter would require us to treat f" as a function of h. However, by treating f as fixed instead (without backpropagating through it), we significantly simplify the FAIR- RET'S implementation. The optimization in step (i) can then be solved generically using specialized libraries such as cvxpy (Agrawal al., 2018 Diamond & Boyd 2016). In our experiments, we find that only 10 optimization steps is enough to get reasonable approximation of the solution. We C. discuss this approximation and visualize each projection f" in Appendix C.T and C.2 respectively.
Figure [ shows that the gradients of the projection FAIRRETS increase with higher values of h. We hypothesize this occurs when y yk; h) > 7hh) because yk; h) is more easily decreased by reducing higher h values than lower ones. Conversely, when y(k; h) y(h), there is more to gain from increasing lower values than higher ones. The sharp bend of the gradients of the DSED SSDD-projcction is explained in Appendix C.2 through an analysis of the projected distributions.
3.3 ANALYSIS
Proposition 2. All FAIRRETS presented in this paper (i.e. Def. 7
8
9
10 and 71
are strict.
Hence, all proposed FAIRRETS can indeed be properly regarded as quantifications of unfairness.
They are differentiable with respect to ht Violation FAIRRETS owe this to the differentiabillty of and projection FAIRRETS the differentiability of D. Hence, FAIRRETS are easily implemented with an automatic differentiation library like Py Torch. The computational overhead is unaffected by the
complexity of the parameters 0 of h, as the gradients VoLy = of both loss functions in the joint objective (Eq. share the computation of the Jacobian ah
at: ae
V and VoR, [ah [0
V,R
is common to minimize Ly using mini-batches; the same batches can be used to minimize Ry. Indeed, this is done in our experiments. Though this makes FAIRRETS scalable, insufficient batch

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sizes will lead poor approximations of the statistics y. Clearly, the mean violation v(h) in viola- tion FAIRRET (Eq.9) computed over mini-batcher batches is not an unbiased estimate of the actual violation over all data. We report the mean SmoothMax loss for increasing batch sizes in Appendix Cs
4 EXPERIMENTS
4.1 SETUP
Experiments were conducted the Bank (Moro al. 2014), CreditCard (Ych & hui Lien 2009), LawSchoof and ACSIncome (Ding et al., 2021) datasets. Each has multiple sensitive features, including some continuous. The classifier / was fully connected neural net with hidden layers of sizes [256, 128, 32] followed by a sigmoid and did not take sensitive features S as input. We trained with all FAIRRETS discussed in Sec. 3 but only report results of Norm, Dis-projection and D D in Appendix C.4 to avoid clutter here. The remaining FAIRRETS, SmoothMax and DKL- cou-projection, were representative for their archetype. These are compared against three baselines implemented in the Fair Fairness Benchmark (FFB) by |Han et al. (202), as their implementation provides these baselines as loss terms in idiomatic PyTorch. They are PRemover (Kamishima al.) 2012). HSIC (Pérez-Suay et al.) 2017). and AdvDebias (Adel et al.] 2019) (where the reverse of the adversary's ' loss is the fairness loss term). In contrast to the FAIRRET implementations, they only accept single, categorical sensitive attribute. Each FAIRRET and FFB fairesss loss was added to the cross-entropy loss in the objective (Eq. 8) in a separate training run for a range of strengths a > O.
We measured fairness over the four statistics y in Table I that relate to Demographic Parity (DP), Equal Opportunity (EO), Predictive Parity (PP), and Treatment Equality (TE) respectively. Violation of each fairness notion is computed maxę V4 (h) (see Eq. (9. Each FAIRRET was minimized with respect to each in a separate training run (and only the optimized violation is reported). The three FFB baselines only consider one fairesss notion, which is to maximize independence between the model's 's output and the sensitive attributes. Their violation is reported for each statistic y.
In summary, there was an experiment run for each dataset, fairness method, fairness strength and statistic y (except for the FFB baselines). Finally, we also use the Unfair baseline with a = 0. Each of these combinations was repeated across 10 random seeds with each different train/test splits.
4.2 RESULTS
Test results are visualized in Fig. train set results are found in Appendix C.5 (and display the same trends). We separately discuss the notions with linear and with
statistics.
For DP and EO, which have linear statistics, both the SmoothMax and DKL- projeccion FAIRRETS are effectively used to minimize the faimees violation with respect to multiple sensitive attributes while minimally suffering a loss AUROC scores, though the projection FAIRRET clearly performs better than the violation-based SmoothMax FAIRRET. As expected, the FFB baselines perform worse than the methods implemented in our FAIRRET framework, since they cannot be configured to optimize the same general range of faireess definitions. Also. their implementation only minimizes bias with respect to a single sensitive attribute, and so they are oblivious to some of the components in S that the violation in Fig. 2 Appendix C.6) though the FAIRRETS still outperform them there as well.
measures. We report their violations this single attribute in
For PP and TE, which have finear-fractions statistics, all methods appear to struggle far more. MmoothMax is most consistent and never makes the fairness violation worse, yet the DKL. -ro-ppooettioon in most cases makes both the fairness violation and the AUROC worse. The same occurs for the FFB baselines. To some extent, this can be attributed to overfitting, as SmoothMax leads to a significantly more consistent reduction of the train set fairness violation than the test set (see Appendix C.5). Still, non-linear fairness notions are clearly harder to optimize, which aligns with the results of Celis et al. (2019). Though |Barocas et al. (2019) conclude that sufficiency (a notion related PP) 'often comes for free', further work is needed to better understand how such notions can be consistently achieved.
4Curated and published by the SEAPHE project
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SmoothMax
Dku-projection
HSIC
PRemover
AdvDebias
Unfair
AUROU
AUROU
AUROC
AUROC
0.92
0.73
0.75
8686
0.40
DP viol8.on
0.80
0.60
0.30
0.10
0.90
0.20
0.05
0.10
0.00
0.30
0.20
B0 vioia.oo
0.15
0.50
0.30
0.20
0.10
0.20
0.06
0.00
11.00
0.00
PP vilatiee
0.20
0.200
0.20
0.10
0.10
0.20
0.00
0.30
0.80
TE violatie
0.20
2.00
0.10
0.10
0.20
0.00
0.00
0.00
0.00
(a) Bank
(b) CreditCard
(c) LawSchool
ACSIncome
Figure 2: Mean test set results with confidence ellipse for the standard error. Each marker
separate combination of dataset, FAIRRET, FAIRRET strength, and statistic. Results in the lower right are optimal. Failed runs (with an AUROC far worse than the rest) are omitted.
CONCLUSION
The FAIRRET framework allows for wide range of faimness definitions by comparing linear- fractional statistics for each sensitive feature. We implement several FAIRRETS and show how they easily integrated in existing machine learning pipelines utilizing automatic differentiation.
Empirically, violation FAIRRETS like SmoothMax consistently lead to trade-off's between fairness and AUROC, though the more involved projection FAIRRETS like the DKL -rojeecent clearly outper- form them on faimess definitions with linear statistics. However, all methods struggle with fairness statistics like PP and TE, which have mostly been ignored in prior
notions that have work. This signals a lucrative direction for future research.

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ETHICS STATEMENT
The FAIRRET framework was made technical tool to help unveil and address mathematical formalization of fairness in machine learning systems. However, such tools should never be con- sidered sufficient solution to truly achieve faimess in real-world decision processes because the social, human component of fairness is completely outside the control of this framework (Selbst et al. 2019). There is significant risk that technologies such ours may anyway be abused to suggest discriminatory bias has been 'removed" from a decision process without actually addressing underlying injustices (Hoffmann) 2019
REPRODUCIBILITY
All proofs, i.e. for Table [ Prop. and Prop. 2 are found in the Appendix A Appendix B contains additional context for Remarks I a Appendix provides experiments referred to Sec. 3.2 a visualization of the projections of the projection FAIRRETS and an empirical assessment of their approximation for fewer optimization iterations. It also evaluates the mean SmoothMax loss for smaller batch sizes mentioned in Sec. 3.3 Furthermore, Appendix exttends the main experiment results of Sec. 4.2 by providing the metrics of the other FAIRRETS, the train set results and fairness violations computed for only a single sensitive attribute. Finally, Appendix D further explains the experiment setup already summarized in Sec. 4.1 i.e. the datasets, hyperparameters, the baselines implementation, the computation of the confidence ellipses and runtimes.
The code for our full experiment pipeline found in the rest of the supplementary material.
The strcamlined package is available at Examples of the code use (at the time of writing the paper) are provided in Appendix
com/ aada-iddaaa comm/aidaa-uen//aaaaaaeeeee fairrret
ACKNOWLEDGGEMENT
The research leading to these results has received funding from the Special Research Fund (BOF) of Ghent University (BOF20/DOC/444 and BOF20/IBF/17 from the Flemish Government under the "Onderzeekspnogramma. Artificiële Intelligentie (AI) Vlaanderen" programme, and from the FWO (project no. GOP9816N, 3G042220, G073924N).
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A PROOFS
A.1
TABLE
Here, we show how the ap. 01, Po. and B1 functions are derived for each of the fairness notions in Table These fairness notions typically defined in the partition faimess setting, and will make the same assumptions here (see Def. D.. Before discussing each fairness notion separately, we make some general observations.
First, there may be a concern that our fairness constraint, i.c. Vk E [ds] y(k;f) = f(f), requires each group's 's statistic y(k; f) to equal the overall statistic y(f) a y(1; f), whereas popular surveys of fairness definitions such as by Verma & Rubin (2018) instead only require that each group's statistic is the same, i.e.
Vk. E [d,] y(k; y(:f))
(A.1)
However, these constraints are equivalent.
Proposition A.1. In partition fairness, with y T:
Vk E [d,] : f):
Vk, E [d,] y(:: f(l
(A.2)
Proof. The forward (=) ) relation is trivial: all group statistics y(k;f) = =(()) then necessarily they are also all equal.
The reverse ( = is less straightforward. Clearly,
Vk, E [d.] : f(;; = f(::f) = Ee ER: Vk [d,] y(k;f) =c
(A.3)
thus suffices to show c = 7(J) in such cases. Let go(X. ax(X.) Y) [(X)Bo(X,) Y) and gi (X,Y) a (X. ((X) y). Then
E S X. Y)
y(k
B[ (X. Y)]
ΣX.Y. Sego (X, S),(X,
Σx,Y, Segi (X,Y))(XXY
Σx.Y 80 (X. 1)
(A.4)
Σx,Y 81 (X.Y))(X.Y,,5+
Σx,Y | 1)
Σx.Y (XX))) Sx
[X, Y) Sx=1] =]
(X, Y) S4 1]]
where the third line used the partition faimnsss assumptions.
Thus, Eq. A.3) also entails that ([gn Y) S = 1] = (X,) Sx = 1].
By the law of total expectation, now have
70 (X. Y)]
[gi (X, Y)]
[[ Y) []]
[(E() |S]
S]]
Σ. Blgo St P] 1)
(A.5)
Σ. Elgi S2 = P] 1)
ΣL EecB(gi(X,)) Sx = 1] 1)
Σ. ((gi()))
1]P(SA=1
=
where the third line again used partition faireess. We thus indeed always have c = =(f). Applying Eq. (A.3) then completes the proof.
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A second observation used derive Table is that faimess notions tend to be defined over proba- bilities involving binary variables. To formulate fairness notions in terms of expectations, we can thus extensively use these binary variables indicator functions. For example, the frequency of true positives P((()) = 1,Y = 1) is easily counted as B[f(X)r).
These observations can be applied to all faimess definitions presented by Verma & Rubin (018)) in their Sections 1 and 3.2 and we refer to their work for detailed interpretation of each definition.
We now show how Table I can be constructed by presenting the statistic of each fairness definition and (a simplified form of) their y function under our notation.
demographic parity (a.k.a statistical parity) (Dwork et al. 2012) was already discussed in Example It requires equal positive rates P(f(X) = 1 Sx = i.e. y(k: = B|Sa f(X)
B[S.
conditional demographic parity (a.k.a conditional statistical parity) (Corbett-Davee Davies et al., 2017, Wachter et al. 2020) conditions demographic parity on a part the input. This is typically an input feature with respect to which we are allowed to discriminate. Let R (0, }} an arbitrary function with a binary output. Then the fairness definition requires equal PX((X) = 1 5(X), Sx = 1), i.e. f(; = [[S]C(X]] statistic is easily extended to also allow non- non-binary č functions.
((SC/( Note that this
equal opportunity (Hardt et al. 2016) was already discussed in Example considers positive rates over only the positive samples (Y 1). It thus compares false negative rates, or equivalently true positive rates or recall P(f(X) = 1 y 1,5x = 1), ie.
f(k;f) ([S,f(X) B[S,Y]
false positive parity (a.k.a. predictive equality) (Hardt al. 2016 Corbett- Corbett-Davies al., 2017) similar to equal opportunity, but for the negative class. Hence, it compares true y 0, S 1), i.e. y(k: [(X)((X) If combined with equal opportunity, enforces equalized odds
negative rates, equivalently false positive rates P(((X)
(Hardt et al. 2016).
already discussed in Example 3. It compares the positive predictive value, a.k.a the precision P(Y = f(X) = 1.Sx = 1), fk:
predictive parity (Chouldechova 2017)
[())
[[S]
false omission parity is similar predictive parity, but for the negative class. It is explicitly discussed in (Verma & Rubin 2018). yet it clearly compares false omission rates
P(Y 1 f(X) S, 1), i.c. y(k; - (1- [[S,(I-
accuracy equality (Berk al., 2021) compares accuracies P(Y = f(X) Sx 1) across groups. Hence, it computes the relative amount of true positives and true negatives of each group, i.e. ykk f) [[SeL/(X))++ (1- ((x)))))))
3[S,]
treatment equality (Berk al., 2021) was already discussed in Example 4. It requires
the fraction of false negatives over false positives to be equal (or vice versa), and therefore does not represent a probability. Its statistic is thus ffk; (1-
A.2 PROPOSITION 1
Proof. Considering the definition of linear-fractional statistics y e Tir Def. 2 we define the shorthand notations cxX, Y)c) (x. () can (X,Y) and B(X, c) = Bo(X, y) cBi (X, Is then straightforward to see that
y(k =c
(X, Y)]
(A.6)
(X Y,c) + f())((X)) Y,,
c))] 0
where the last step uses the linearity of the expectation operator E. The resulting constraint is linear with respect to since and B both functions of functions that do not depend on f.
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A.3 PROPOSITION 2
To show the strictness of the proposed FAIRRETS, we use a separate strategy for violation and pro- jection FAIRRETS.
A.3.1 VIOLATION FAIRRETS
Proof. Per Def. 3 it is easily seen that v(h) = = h E T. This also holds in the case where 7(h) = 0, as we can define (h) ly(k; h)] there instead. Hence, we only need to show that (h) 0 = v(h) 0 to show the strictness of violation FAIRRETS Ry.
For the Norm FAIRRET, the strictness is obvious: norm defined over a vector space is always assumed to equal zero only for the vector of zeros, i.e.
(h) [|v()]] =0 = v(h) =.
(A.7)
For the SmoothMax FAIRRET, the strictness follows from the log ds adjustment term:
(h) = log ke[d,]
Σ
exp(vx(h)) log d, 0
Σ
exp(vx(h)) d,
(A.8)
kE[d+]
v(h) =0
where the last step follows from the non-negativity of vk-
A.3.2 PROJECTION FAIRRETS
Proof. A projection FAIRRET easily shown to be strict if its divergence measure satisfies
D(f(X) I| h(X) A D(F(X) h(X) =0 f(X) h(X)
(A.9)
Indeed, if h Fy. then also h E (7())) by construction. We can then choose f h and use Eq. (A.9 to get D(F(X) I| h(X)) = 0. This shows h Fy = Ry (+)= =0
Conversely, the assumed properties of D in Eq. (.9 entail
(h) + min (7(1))())
B[D(S()) II 4(X))] =0
fʃ (7(h) VX Rd. f(X) h(X)
(A.10)
h (h))
where the second line assumes the support of X equals Rd. Since (y()()) Fy, have (h) = = h E Fy-
To show the strictness of the projection FAIRRETS, it thus suffices show that the divergences D in
Def. 99 and all satisfy the requirements in Eq. AA.))
It is well-known property of the Kullback-Leibhe divergence DRL. that Eq. A.9 holds Csiszar 1975).
The properties of the DRL- -iveeiieeeee also trivially imply non-negativity for the Jensen-Shannon divergence. Moreover, it entails
(is((()) II ((X)) A (x)( Il m()) + (h(()(X) mX))
2
(A.11)
(xX) || m(X) = (h(X) m()) =0
f(X) m (X) = h(X)
which means Eq.
A.9 indeed also holds for DJs-
Finally, it is clear that DSED satisfies Eq. (A.9), as the Euclidean distance non-negative and only zero for overlapping points.
'If the support of X does not equal Rd, then can simply reformulate our framework only consider function outputs f(X and h(X) for points X that do lie in its support.
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B ADDITIONAL DISCUSSION
B.1 ADDENDUM TO REMARK
Observe that, for y L
fk: _ (X))
(B.1)
and
7() (X) ( B[an X, Y) + (X)))
(B.2)
In Remark 1 we (non-exhaustivelly) mention cases for Sg R where y(k =7O holds:
1. S is deterministic, i.e. Sk for constant E R. Due to the linearity of the expectation operator, we trivially have f(k;f) = 7(f). Though this case is degenerate, we should indeed expect a fairness criterion to hold if the random variable Sg expresses no information about an individual (and as such cannot be grounds for discrimination).
Sk has no linear influence on the numerator or denominator of y, i.e.
cov(Sk, or (X, Y) f(x) Y)) cov 0 (X,Y)+ ((X) (X) =
(B.3)
Indeed, using the definition of the covariance operator, we have
f): = cov(SA, on (X, X, V) cov(S 0r (X, (() f(X)
/ + V) ((X) Y)] B (X, f(x)
(B.4)
Unfortumately, Eq. B33 is only a sufficient condition for y(k: f) y(f) in general. Still, it becomes necessary condition for the DP fairness notion (which always has a1
= cov(Sz, 1)= 0)-
Yet, though we argue y(k; f) = 7() is sensible enforce for continuous variables S3 E R. must be stressed that fineao-fractional statistics check linear effects of S on f(X). Higher-order moments will thus not be measured.
For example, if So denotes 'man-ness' and S₁ denotes 'woman-ness', then individuals who identify with binabinar gender may quantify their sensitive features as So = 50% and S = 50%. However, linear covariance statistics (like ours) will not consider specific discrimination directed at those with, e.g., S = 50%. Instead, this will be taken into account as 'half as influential' compared to individuals who identify as 100%.
B.2 ADDENDUM TO REMARK 2
In Remark we mention that intersections of demographic groups will not be considered in fairness constraints if partition fairness does not hold.
For example. assume we want to check DP fairness with statistic y(:; f) E| (X) for four E[
uniformly selected samples with scores (0.7, 0.3, 0.7, 0.3) given by f. values (1, 0, 0) for sensitive variable So and values (1, O, 0, 1) for sensitive variable Sj. Then y(; = y(1:f) f) 0.5. However,
0.3. Here, the individual with sensitive feature vector S = (0, 0), i.e. at the intersection of So = and S O, thus receives worse scores than the others.
[[ 0.7 and (1(1-S)) ((X)
S|
B.3 ADDENDUM TO REMARK
In our discussion of fairness definitions, we assume that we are enforcing constraints on classifiers Rdd - [0. 1), i.e. classifiers with hard decision boundary. In practice, however, it is common to base decisions off a parameterized regressor Rd, - R, e.g. with r a neural network. then deterministically collect decisions from as f(X) 1,(x)=0 with the
For example,
indicator function. However, such a thresholding function has a gradient of zero with respect to r and discontinuity in r(X) = 0.
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Several works have investigated directly using r(X) in faimess constraints (afarr al. 2019). For example in DP faimess, directly computing y(k; h) will already enforce that the mean scores r(X) should be equal across groups. Though this leads to interesting convex optimization problems for fair r, it has heen noted that this only observes a relaxed form of actual fairness constraints (Lohaus et al. 2020).
middle ground is to construct a probabilistic classifier f as a Bernoulli distribution with parameter h(X) or((X)) where or denotes the logistic function (s)) (1 exp(-s))~ Hence, f(X) is fully specified through PX/(X) = x) h(X). In the definition of y, we can then simply replace the classification function in the expectation by the scoring model h, because [f X]] [[h(X)]. Still, the fact that a probabilistic classifier is now not only dependent on X but also on the randomness involved sampling f(X) introduces some noise into the decision process that we may wish to avoid. There is also the danger that in practice, real-world decisions would still be made according to a hard threshold on the scores ((X), e.g. on f(X) (X))=0.5
If this becomes issue, then we could make use of recent work (Padh et al. 2021 Bendekgey Sudderth) that has investigated surrogate functions that are better approximations of the thresh- olding function 1, than the logistic function a. For example, suitable surrogate is the scaled logistic function (s) (ass), since lima- (s)) 1s00 for s # O. However, note also that g by (.) = Surrogate functions like da thus allow us to trade-off the hardness of the classification with the quality of its gradient.
C ADDITIONAL RESULTS
C.I APPROXIMATE PROJECTION
In Sec. 3.2 suggest that the convex optimization of the projections f" already converge quite well after only 10 iterations. By placing such a limit, we can significantly reduce the computational cost of (h). though Ry (h) will be an overestimate projections f with smaller divergence to h may not have been found yet).
In Fig.
we observe that training with at most 10 iterations per projection is indeed enough to minimize the DP violation while also being much faster to compute. Unexpectedly, we even observe the DP violation be slightly lower after training with most 10 iterations than with more than 10, which could indicate that fewer iterations can have positive effect on the training process.
C.2 PROJECTION VISUALIZATION
In Fig.
visualize the projected distributions for the example in Fig, I The DKL- and Dss- Dis-projectoo projections appear to transform the shape of the probability distribution, whereas the DSED projection appear to linearly shift the probabilities within group.
For the latter, this appears to form a problem, because its behavior leads to a large gap in densities between I and the projection f the edges of the [0, 1] interval. For example for So 1, the scores are linearly shifted to the left (lower probability scores), meaning that no scores are left in the high probability range, and too many are allocated the low probability range. The 'blocking' of this shift on the low end for So I and on the high end for S 1 is in fact the reason for the 'crack' in the gradients of the DEED DSED-Projecto projection in Fig. since the probabilities of h cannot be projected beyond these edges.
C.3 MINI-BATCHING THE FAIRRET
In Sec. 3.3 propose that FAIRRETS Ry can be minimized using the same mini-batches that use to minmiiee the cross-entropy loss Ly. However, batches clearly need be large enough to adequately represent the imbalances for all sensitive features S. Hence, report experiment in Fig. where we take an unfair classifier and compute the mean SmoothMax loss over batch sizes with increasing granularity.
From a batch size of approximately 1024, the mean loss over all batches already closely matches the Max loss computed over the full test set (39 133 samples). Note that for very small batch
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Figure C.I: (left) Test set DP violation, with a similar experiment setup as Fig. Zon the ACSIncome dataset. Each result bar results from separate training run with the -roojeooett fairret that was minimizing the DP violation. The configurations only differ in the maximum number of iterations used in the convex optimizations that compute the actual Pxo-ppooestsoon f" (see Sec. 3.2) (right) The total training time of these runs with standard error.
sizes, the loss becomes mostly meaningless, batches will even contain all members of each protected group. To be on the safe side, we use batch size of 4096 in all our experiments.
C.4 OTHER FAIRRET RESULTS
The test set results on the Norm, Dis-projection and DSED-Projectoo FAIRRETS are shown in Fig. C.4 The Norm FAIRRET follows very similar gradient as the SmoothMax FAIRRET, whereas the D Dis-projection and DSED -roojproeet give similar results as the D
C.5 TRAIN SET RESULTS
In Fig. C.5 we show the train set results for our main experiments. These follow the same trends results, though with higher AUROC scores and lower fairness violations. The most important difference is the performance of the SmoothMax fairret, e.g. on the CreditCard dataset, which obtains significantly lower fairness violation for the difficult PP and TE faimess notions than it did on the test set.
as the
C.6 FAIRNESS VIOLATIONS FOR SINGLE SENSITIVE ATTRIBUTE
The FFB implementation (and indeed most fairness tools) only mitigate bias with respect to single, categorical sensitive feature. our datasets (see See. D.T, these were the following:
Bank: marital status
CreditCard:
LawSchool: sex
ACSIncome: sex
The fairesss violations for these single features are shown in Fig. C.6 Importantly, the FAIRRET experiments redone by training with only this single feature in mind when computing the FAIRRET loss. Even for the DP fairness notion, the FAIRRETS remain competitive and the DKL.- projection obtains the best trade-off.
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of Dru-projection
of Dos-projection
of Dssn-projoction
individuals with Sq
individuals with St
Donsity
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Probabilty Score
Figure C.2: Starting from the same setup as in Fig. we show the probability scores both h (full line) and the projected distributions f of each projection FAIRRET (dotted lines). The y-axis shows the KDE densities of these scores, all on the same scale.
D ADDITIONAL EXPERIMENT DETAILS
D. DATASETS
We used four different datasets for the evaluation of our framework, namely the Bank Marketing dataset (Moro et al. 2014), Credit card clients dataset (Ych & hui Lien, 2009). Law School Ad- missions dataset and the ACSIncome dataset from Folktables (Ding et al. 2021). Their main advantages their range of sensitive attributes, their recency and their curation quality.
We deviate from the normal practice of using the German Credit and Adult data sets as advised by Fabris al. (Fabris al., 2022) due to the "contrived prediction tasks, noisy data, severe coding mistakes, limitations in encoding sensitive attributes, and age" of these datasets.
D.1.1 BANK
The Bank marketing dataset was collected by a Portuguese bank between May 2008 and June 2013. The dataset includes all information the bank has client, information about the previous attempt for the client to subscribe for long-term deposit, some economic information and if the client decided go for long-term deposit following the telephone call.
CCuaaeed and published by the SEAPHE project
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Figure C.3:
Test set SmoothMax loss (h) with y the positive rate statistic (enforcing the DP notion), computed for an unfair model h trained with the same setup in Fig. 0 Each loss is computed over the entire test dataset, but chunked using different batch sizes. For smaller batch sizes, the mean SmoothMax loss is an overestimate of the actual SmoothMax loss computed over all 39 133 samples.
The dataset itself contains the information 41 108 telephone contacts. Important to note for this dataset is that the outcomes are severely unbalanced with only 4 640 of outcomes belonging the positive class.
The sensitive attributes which were used such in this dataset the age and marital status of person. A person's i eduction was not included as a sensitive attribute as discriminating on that value could arguably be justifiable in this situation.
During preprocessing, we dropped five features: three relating to the outcome the previous mar- keting campaign as often there date when the current marketing call took place. If the value of certain features were unknown then they were were mapped onto the 'False' value, except for the case if the marital status of the person was unknown. In that instance the row was simply dropped, since this only occurred for 80 samples the entire dataset.
record of previous contact and two features relating the
D.1.2 CREDITCARD
The Credit Card clients dataset is data from bank in Taiwan in October 2005. The goal predict whether client would default their credit in the next month. The features include the allocated credit the client, personal information, the status of previous payments, amounts of previous payments and the amounts on previous bill statements.
In total the dataset contains 30 000 records. In 5529 of these instances the client defaulted in the next month.
This dataset contains a wide range of sensitive attributes, namely sex, education, marital status and age.
total of 522 samples were removed from the dataset as they contained values unspecified in the documentation of the data or if their eduction status fell under the category others. That was mainly done as only 123 samples were of this category, making it too small to maintain adequate statistical power for the faimess measure of the group
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Figure C.4: Test set results for the experiments in Fig.
but with different FAIRRETS.
D.1.3 LAWSCHOOL
The Law School admissions dataset contains information about whether a student was accepted to the law school, which we tried to predict in our experiments. The dataset also contains the student's s race, gender, whether they live where they applied to law school, what college they applied to, the year they did this, and their LSAT and GPA scores.
Although this dataset is hand-curated by the SEAPHE project, it is still fairly sizeable with 65 535 samples. Only 24.84% of these samples had positive outcome.
The dataset only contains two sensitive attributes: race and sex. This is the only dataset where the age features are not available. Interestingly, the range of values for the race attribute is fairly balanced across all races, except for white which is overly represented with 39 742 samples.
In the preprocessing step only the year of the application column not used feature.
1.1.4 ACSINCOME
The ACSIncome dataset has the familiar goal of predicting whether an individual's annual income is above $50 000. The source of the data is the US census study. This only includes individuals above the age of 16, who indicate having worked at least one hour per week and reported an income above $100. A myriad of personal information is available in the dataset.
The dataset is significantly larger than the others used, with 195 665 samples. also more bal- anced as it has 80 335 positive samples.
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Figure C.5: The train set results corresponding with Fig.
Four sensitive attributes are used for the calculations. In this case age, marital status, sex and race. Information about employment and education is not included as sensitive attributes.
Due to the large amount of race groups in the survey, it is necessary to simplify this trait in order to have each group be at least a total of 1% of the data set to guarantee statistical significance for each group.
D.2 HYPERPARAMETERS
In addition to the hyperparameters discussed in Sec. 4.1 we also mention that our model was op- timized with the Adam optimizer implementation of Porcor,, with learning rate of 0.001 and batch size of 4096. The loss was minimized over 100 epochs, with 2 = O for the first 20 to avoid constraining h before learns anything.
To find these hyperparameters, we took the 80%/20% train/test split already generated for each seed, and further divided the train into a smaller train and validation with relative sizes 80% and 20% respectively. Keeping the FAIRRET strength 2 O, we performed grid search on the neural architecture range
[[]. [128], [128, 32], [128, 64, 64,32], 32], [256, 128, 32], [512, 256, 128, 32], [512, 512, 256, 128, 32]] (D.1)
in combination with the learning rate in range [0.01, 0.005, 0.001, 0.0005, 0.0001]. We then selected the combination with the best AUROC the validation set of the Bank dataset. The number of epochs and the batch size were tuned manually based on the convergence properties of the validation AUROC.
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Figure C.6: Test set results for the experiments in Fig. but with fairness violations computed for single sensitive feature. For these results, FAIRRET experiments were redone and optimized for only this feature specifically.
D.3 FFB BASELINES
For all FFB (Han et al. 2023 baselines, we used the publicly available implementation with only minor adjustments make them fit in experiment pipeline.
From their implementation, we used the following baselines.
HSIC minimizes the fillert-Schhiie Independence Criterion between the model's predic- tion probabilities and the sensitive attributes (Pérez-Suay et al. 2017).
PRemover minimizes the mutual information between the model's prediction probabilities and the sensitive attributes (Kamishima et al. 2012)..
AdvDebia maximizes an adversary's 's cross entropy between its predictions for the sensitive attribute and the actual sensitive attribute, given the last hidden layer of A|Adel et al. (019)) after trying several configurations, we achieved the most stable results with an adversarial net of one hidden layer of size 32.
We again stress these implementations only optimize for the specific fairness notion listed above and only for single categorical sensitive attribute.
benchma rk committ///////////mm/ccm/.......
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At the time of writing, the FFB also contains implementations for regularization term that mini- mizes the gap to obtain DP and EO, but this is highly similar to our violation fairrets and would not be an informative comparison.
D.4 CONFIDENCE ELLIPSES
The confidence ellipses we use in Fig. literature. Yet, they work well for purposc of comparing trade-off's between metrics that may be noisy depending on randomness during training and dataset split selection.
Fig.
C.4 and Fig. C.5 are uncommon in machine learning
Recall that 1-dimensional confidence intervals typically assume mean estimator to be normally distributed. The confidence interval then denotes the uncertainty of the sample mean using the standard error. Similarly, confidence ellipses assume 2-dimensional point, i.e. the 2-dimensional mean estimator, to have a multivariate normal distribution that can be characterized through the sample mean and standard error statisties.
Our implementation of the confidence ellipses follows featured implementation on matplot 1 1 However, crucial difference is that this implementation computes confidence interval for 2-dimensional random variable based on the covariance matrix for the standard devi- ation of samples of that variable. Following observations by Schubert and Kirchner (Schubert & Kirchner (2014), we instead want to show the uncertainty of the mean estimator, which should use standard deviation that estimator, i.e. the covariance for the standard error. This is accom- plished by dividing the covariance matrix in the ma matplot! ott ib implementation by the number of seeds (5) we use our experiments.
D.5 COMPUTATION COST AND RUNTIMES
We report some of the runtimes in Fig. D.I that illustrate the difference in computational cost be- tween FAIRRETS and baselines during the main experiments of Sec. Note that both our FAIRRET implementation and the FFB implementation were designed for intuitive use in research and not yet optimized for runtime speed.
Norm Max
DEL- projection
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Figure D.I: Runtimes for the ACSIncome dataset experiments discussed in Sec. 4 with strength A (except for the Unfair baseline). The FAIRRETS were optimizing the DP fairness notion. Note the log scale.
All experiments in Sec. 4 were conducted internal server equipped with 12 Core Intel(R) Xeon(R) Gold processor and 256 GB of RAM. All experiments, including preliminary and failed experiments, cost approximately 100 hours per CPU.
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import torch
import torch.nn. functional as
From fairret.statisst import VruePositivekate from fairret. loss. violatioii import Normloss
The TruePositiveRate class is a subclass of Linerrrraction statistic TruePositiveRated
11 The fairret modules accept any iincafFracttonnassttaiiiioo instance.
12 fairret NormLoss (statistic)
13 fairret_strengee = 1.0
16 def train_epoch (train Loaddenn_ood,, model, optimizer)
17
for feat, sens, target in srain_Loaded optimizer.zero_graan
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legit model (feat)
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bce loss F.binary_crreaee cross entropy_ with with_logits (logit, target)
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fairret_loss = fairret (logit, feat, sens, target)
loss = bce
bce_loss loss + fairret starrettsttength * fairret loss (oss.backward()
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optimizer.sten
Listing E.1: Example use of the FAIRRET library in simple PyTorch setup.
E CODE USE EXAMPLES
Listing E.1 displays code example of how the FAIRRET easily be deployed in typical PyTorch (Paszke et al.) 2019) setup. It suffices to simply load a subclass of LinearrractionaStaate
and pass on FAIRRET implementation instance such as NormLoss (as defined in Def. 7. The FAIRRET is then used to compute the quantification of unfaimesss as a loss like any other PyTorch. In this case, we use the true positive rate statistic to pursue the faimess notion of equalized opportunity (EO).
In Listing E.2 we provide an example implementation of LinearFracte itw.Fraatioaaltttttstttittt. class, which only entails the specification of the and ß functions as in Table [
denccc Liccee__ell__ ipse.
html
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import torch
from fairret.statiste import Lihearfraattoonasstattisiti
class Accuracy Lnorrararrrttiinnstttsssit... name="ace");
alpha_0
def nom _om_intercept. (self, feat, label) return label
11
beta_0
12
def nem_slope (self, feat, label): return 2 label -
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alpha_l
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def denom_intercept (self, feat, label):
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return torch. ones
beta_
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def Sencm_slopel (self, feat, label) :
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return forch.zeros (feat shape [0])
Listing E.2: Example implementation of a custom LinearFractionnlStttiiii
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Reviews:
[{"review_id": "CSeaIq9KZ7", "reviewer": "Reviewer_ALkR", "summary": "This paper presents a tool that implemented generalized group fairness metrics that can be used in automatic differentiation libraries. The fairness metrics are expressed as a linear-fractional statistic, which can be further represented as a smoothed regularization term. The authors also present an alternative projection method that penalize the divergence between models. Through extensive experiments, the authors shows the framework is lucrative for the optimization of fairness constraints.", "questions": "1. If the model $h$ is not a probabilistic classifier, then Equation (4) is no longer differentiable. Can the framework still be useful? \n2. How is the projection $f^\\star$ initialized in the constrained optimization problem of Equation (5)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The presented SmoothMax regularization terms are elegant and provide expressive representations for widely applied group fairness metrics.\n- The methods can be combined with automatic differentiation tools, such as PyTorch.\n- The methods can be naturally applied with multiple axes of sensitive attributes, allowing wider applications.", "weaknesses": "- The method still applies relaxed fairness metrics, rather than the exact metrics as the regularization terms.\n- A superior learning objective should be a minimax game with the optimization of the $\\lambda$-player. As far as my understanding, the authors use fixed $\\lambda$ values as a hyper-parameter and run grid search to get the optimal results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TT8A7gNDKB", "reviewer": "Reviewer_eBDd", "summary": "The authors present a general framework for formulating fairness regularization terms which are differentiable. Their framework encompasses a wide range of group fairness notions. The authors then propose a series of regularizers for enforcing the fairness definitions. Finally, an experimental evaluation is given.", "questions": "Some minor questions:\n- Why do you call your method \"Partition Fairness\" and not \"Group Fairness\" as is common in the literature?\n- Regarding continuous sensitive variables. It seems that this approach only captures linear correlation between the sensitive variables and the score. Is this correct?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I think the authors propose a valuable contribution to the fairness community. Specifically, I appreciate the effort the authors make on combining multiple fairness notions into a general framework. I also think that providing a python package can be valuable for research and adaptation of fairness in ML.", "weaknesses": "The theoretical contribution of this work is in my opinion quite limited. While combining different fairness notions is valuable, I do not think that this is in itself a theoretical contribution. For instance, if in (3) you fix $\\overline{\\gamma}(h)=c$, then a norm based regularizer would be convex in $f(X)$. Thus if the model is linear, you would get a convex regularizer. Combine this with a convex $\\mathcal{L}_Y$ and your problem is convex in the model parameters. This would allow you to get fast convergence rates.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AOynSDvpnx", "reviewer": "Reviewer_hHUL", "summary": "In this paper, the authors study the problem of fairness in ML. To be specific, they introduce a formal framework which facilitates defining regularization terms for fairness which can be minimized using auto-differentiation tools. The authors report significant improvements over baselines.", "questions": "Please see Weaknesses.\n\n**After Rebuttal**\n\nI've read the comments provided by other reviewers and the responses by the authors. I find that the authors have sufficiently addressed my concerns. Looking at the sample code in Appendix E and the implementations in FFB, I see the contribution of the paper better. I think it will be beneficial for the community. Therefore, I've changed my recommendation.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+ The paper provides a good formal coverage of the fundamental concepts.", "weaknesses": "1. I have concerns about the novelty. It appears that the paper's novelty is a formal fairness regularization framework. The paper criticizes existing frameworks for not being formal and limited in terms of fairness definitions. However, the paper does not showcase what the benefit of this formal framework is. Moreover, it is not clear why FFB or FairTorch cannot be extended to include more fairness definitions.\n \n2. I find \"fairness tools\" misleading. This is sometimes used to refer to fairness measures,  sometimes to their approximations as regularization terms and sometimes to their implementations.\n\n\nMinor comments:\n- I would recommend following the following guide while writing equations: \nhttp://www.ai.mit.edu/courses/6.899/papers/mermin.pdf", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DQNHySyicJ", "reviewer": "Reviewer_BABL", "summary": "Paper claims that current tools for machine learning fairness only admit a limited range of fairness definitions and have seen little integration with automatic differentiation libraries.\nFor this reason they introduce a framework of fairness regularization terms which quantify bias as modular objectives that are easily integrated in automatic differentiation pipelines.", "questions": "There are plenty of work that proposes differentiable (and in some case even convex) regularisers (e.g. [1] but in ICML, NeurIPS, etc. you can find plenty of work on this). Paper should elaborate on that.\n\n[1] Exploiting MMD and Sinkhorn Divergences for Fair and Transferable Representation Learning, NeurIPS 2020.\n[2] Deep Fair Models for Complex Data: Graphs Labeling and Explainable Face Recognition, Neurocomputing 2021.", "limitations": "", "rating": 3, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "The overview, FAIRRET, and results are interesting and valuable.", "weaknesses": "Authors claim is an overstatement in fact there are plenty of work that proposes differentiable (and in some case even convex) regularisers.\nState of the art is largely incomplete.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)