FairGrad: Fairness Aware Gradient Descent
Authors: Gaurav Maheshwari
Michaël Perrot
Venue: Accepted by TMLR ()

Abstract: We address the problem of group fairness in classification, where the objective is to learn models that do not unjustly discriminate against subgroups of the population. Most existing approaches are limited to simple binary tasks or involve difficult to implement training mechanisms which reduces their practical applicability. In this paper, we propose FairGrad, a method to enforce fairness based on a re-weighting scheme that iteratively learns group specific weights based on whether they are advantaged or not. FairGrad is easy to implement, accommodates various standard fairness definitions, and comes with minimal overhead. Furthermore, we show that it is competitive with standard baselines over various datasets including ones used in natural language processing and computer vision.

FairGrad is available as a PyPI package at - https://pypi.org/project/fairgrad

Published in Transactions on Machine Learning Research (08/2023)
FairGrad: Fairness Aware Gradient Descent
Gaurav Maheshwari
gourae.maahssuuuriihnnrri.
Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France
Michaël Perrot
michuel..erroo@@nrii.
Unio. Lille, Inria, CNRS, Centrale Litle, UMR 9189 CRISIAL, F-59000 Lille, France
Reviewed on OpenReview: htlps:// https///ppenreveew.neoe net/ forum id= Of81
Abstract
We address the problem of group fairness in classification, where the objective is to learn models that do not unjustly discriminate against subgroups of the population. Most exist- ing approaches limited to simple binary tasks involve difficult to implement training mechanisms which reduces their practical applicability. In this paper, we propose Fair- Grad, method to enforce fairness based on a re-weighting scheme that iteratively learns group specific weights based on whether they are advantaged or not. FairGrad is easy to implement, accommodates various standard fairness definitions, and comes with minimal overhead. Furthermore, we show that it competitive with standard baselines over various datasets including ones used in natural language processing and computer vision.
FairGrad available as PyPI package at httpps//pppp.rrgppooccc//// ffairgrrd
1 Introduction
Fair Machine Learning addresses the problem learning models that are free of any discriminatory behavior against subset of the population. For instance, consider a company developing a model predict whether person would be suitable hire based on their biography. A possible source of discrimination here can be if, in the data, available to the company, individuals that part of subgroup formed based their gender, ethnicily, other sensitive altributes, are consistently labelled as unsuitable hires regardless of their true competency due to historical bias. This kind of discrimination can be measured by a fairness notion called Demographic Parity (Calders et al. 2009). the data is unbiased, another source of discrimination may be the model itself that consistently mislabels the competent individuals of a subgroup as unsuitable hires. This can be measured by a fairness notion called Equality of Opportunity (Hardt et al. 2016).
Several such fairness notions have been proposed in the litcrature different problems call for different measures. They can be divided into two major paradigms, namely (i) Individual Fairness (Dwork et al. 2012 Kusner et al. 2017) where the idea is to treat similar individuals similarly regardless of the sensitive group they belong to, and (ii) Group Fairness (Calders et al.) 2009: Hardt et al. 2016 Zafar et al., 2017a Denis et al. 2021b where the underlying idea is that no sensitive group should be disadvantaged compared to the overall reference population. In this paper, we focus on group fairness in the context of classification where we only assume access to the scnsitive attributes during the training phase.
The existing approaches for group fairness in Machine Learning may be divided into three main paradigms. First, pre- preprocesssing methods aim at modifying dataset to remove any intrinsic unfairness that may exist in the examples. The underlying idea is that a model Icarned on this modified data is more likely to be lair (Dwork et al. 2012 Kamiran & Calders 2012 Zemel el. al. 2013 Feldman et al.) 2015 Calmon et. al. 2017). Then, becomes fair (Kamiran et al. 2010 Hardt et al. 2016 Woodworth al.] 2017 Iosifidis al. 2019 Chzhen et al. 2019). Finally, inocescssinng methods aim at learning model that fair and accurate a single
approaches modify the predictions of an accurate but unfair model so that it
step (Calders & Verwer 2010 Kamishima et al. 2012 Goh et al. 2016 Zafar et al. 2017a b Donini et. al.

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# The lsbrary ss aveilable at from fairgrad.torch import CrossEntropylooss
# Bame as PyTorch's loss with some additional meta data. # A fatrness rate of 0.01 is good rule of thund for standardized data. criterion CromsBntropyLoosyyyerra traain, s_train, fairness_measure, fairness fairness_rate=O.00 rrae==.00))
The dataloader and model are defined and used the standard way.
for x, y, data_loaded optimizer wxiezeerrrrr zero__rad(
loss criterion (x), y, s)
loss. ()
(ptim)
Figure 1: A standard training loop where the PyTorrch's loss is replaced by FairGrad's loss.
2018
Krasanakis et al. 2018 Agarwal et al. 2018 Wu et al. 2019:
Cotter et al., 2019
Iosifidis & Ntoutsi
2019 Jiang & Nachum 2020: Lohaus et al. 2020 Roh et al. 2020
Ozdayi et al., 2021). In this paper, we
propose new in- in-processing group fairness approach based as kind of post-processing approach by fine-tuning existing classifiers.
weighting scheme that may also be used
Motivation. In-processing approaches can be further divided into several sub-calegories (Caton & Haas 2020). Common amongst them are methods that cast the fairness task as a constrained optimization problem and then relax the fairness constraints under consideration to simplify the lcarning process (Zafar et al., 2017a; Donini al.] 2018: Wu et al.) 2019). Indeed, standard fairness notions are usually difficult to handle due to their non-convexity and non-diiferenttiabliyy Unfortunately, these relaxations may be far from the actual fairness measures leading to sub-optimal models (Lohaus et a.l. 2020). Similarly, several approaches address the fairness problem by designing specific algorithms and solvers. This is, for example, done by reducing the optimization procedure to a simpler problem (Agarwal el al. 2018). altering the underlying solver (Cotter et al. 2019), or using adversarial learning (Raff & Sylvester 2018). However, these approaches are difficult to adapt to existing systems as they require special training procedures or changes in the model. They are also limited in the range of problems to which they can be applied. For example, the work of AAgarwal et al.] (2018) can only be applied in binary classification setting, while the work of Ozdayi ct al. (2021) is limited to two sensitive groups. Furthermore, they may come with several hyper-paramoters that need to be carefully tuned to obtain fair models. For instance, the scaling parameter in adversarial learning (Raff& & Sylvester 2018 Li et al. 2018) or the number of iterations in inner optimization for bi-level optimization based mochanisms OZdayi et al., 2021). The complexity of the existing methods might hinder their deployment in practical settings. Hence, there is a need for simpler methods that are straightforward to integrate into existing training loops.
Contributions. In this paper, we present FairGrad, general purpose approach to enforce fairness in empirical risk minimization solved using gradient descent. We propose to dynamically update the influence of the examples after each gradient descent update to precisely reflect the fairness level of the models obtained at each iteration and guide the optimization process in a relevant direction. Hence, the underlying idea is to use lower weights for examples from advantaged groups than those from disadvantaged groups. Our method is inspired by recent reighttht approaches that also propose to change the importance of each group while lcarning a model (Krasanakis et al. 2018 Tosifidis & Ntoutsi 2019 Jiang & Nachum 2020 Roh et al., 2020; Ozdayi et al., 2021). We discuss these works in Appendix A Interestingly, we also find that FairGrad can be seen solving kind of constrained optimization problem. In Section 2.2 expand upon this link and show how FairGrad can be seen as solution that connects these two kinds of methods.
A key advantage of FairGrad is that it is straightforward to incorporate into standard gradient based solvers that support examples re- weighting like Stochastic Gradient Descent. Hence, we developed a Python library (provided in the supplementary material) where we augmented standard PyTorch losses to accommodate our approach. From a practitioner point of view, means that using FairGrad is as simple as replacing their existing loss from Py PyTorch with our custom loss and passing along some meta data, while the rest of
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the training loop remains identical. This is illustrated in Figure | It is interesting to note that FairGrad only brings one extra hyper-parameder, the fairness rate, besides the usual optimization ones (learning rates, batch size, ...). Moreover, FairGrad incurs minimal computational overhead during training as it relies on objects that are already computed for standard gradient descont, namely the predictions on the current batch and the loss incurred by the model for each example. In particular, the overhead is independent. of the number of parameters of the model. Furthermore, as many in-processing approaches in fairness (Cotter et al. 2019 Roh et al., 2020), FairGrad does not introduce any overhead at test time.
Overall, FairGrad is lightweight fairness solution that is compatible with various group fairness notions, including exact and approximate fairness, can handle both multiple sensitive groups and multiclass problems, and can fine tune existing unfair models. Through extensive experiments, we also show that, in addition to its versatility, FrirGrad is competitive with several standard baselines in fairness on both standard datasets as well as complex natural language processing and computer vision tasks.
2 Problem Setting, Notations, and Related Work
In the remainder of this paper, we assume that we have access to a feature space X, a finite discrete label space y, and set S of values for the sensitive attribute. We further assume that there exists distribution D E Dz where Dz is the set of all distributions over 2 = X' x 3' x S. Our goal is then to learn accurate model he E H. with learnable parameters 0 E R', such that he : X + y is fair with respect to given fairness definition that depends on the sensitive attribute. In Scction 2.1 we formally define the family of lairness measures that are compatible with our approach and provide several examples of popular notions encompassed by our fairness definition.
As usual in machine learning, we will assume that D is unknown and that we only get to observe finite dataset T = { st) of examples drawn i.idd from D. Let IP (E(X, Y, S)) represent the probability that an event E happens with respect to Y,) empirical estimate with respect to T where Ip is the indicator function which is when the property P is verified and 0 otherwise. In the remainder of this paper, all our derivations will be considered in the finite sample setting and we will assume that what was mcasured on our finite sample is sufficiently close to what would be obtained if one had access to the overall distribution. This seems reasonable in light of the previous work on generalization in standard machine learning (Shalev-Shwartz & Ben-David 2014) and the recent work of Woodworth et al. (2017) or Mangold et al. (2022) which show that the kind of fairness measures we consider in this paper tend to generalize well when the hypothesis space is not too complex, as measured respectively by the VC the Nalarajan Dimension (Shalev-Shwartz & Ben-David, 2014). Since these generalization results only rely on a capacity measure of the hypothesis space and are otherwise algorithı agnostie, they are applicable to the models returned by FairGrad when they have finite VC or Natarajan dimensions, This is for example the case for linear models.
D while P (E y. 8))
En= II(anans)))) is an
2.1 Fairness Definition
We assume that the data may be partitioned into K disjoint groups denoted TI....,,Te...., TK such that UK To = 'T and n Th = D. These groups highly depend on the fairness notion under consideration. They might correspond to the usual sensitive groups, as is the case for Accuracy Parity (see Example D or might be subgroups of the usual sensitive groups, as in Equalized Odds where the subgroups are defined with respect to the true labels (see Example 2 in Appendix B For each group, we assume that we have access to a function F, D" x H + R such that Fs > 0 when the group l is advantaged by the given classifier and FR < when the group & disadvantaged. Furthermore, we assume that the magnitude of F3 represents the degree to which the group is (dis)advantaged. Finally, we assume that each F& can be rewritten as follows:
PA ho) = C2 + Σ (ho(t)
(1)
F=1
where the constants C are group specific and independent of heg The probabilities È (ho(2)) # TT)) repre- sent the error rates of by over each group Te with slight abuse of notation. Below, we show that Accuracy
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Parity [afarr et al. 2017a) respects this definition. In Appendix
B we show that Equality of Opportu-
nity (Hardt et al. 2016), Equalized Odds (Hardt et al. 2016), and Demographic Parity (Calders et al. 2009 also respect this defimition. It means that using this generic formulation allows us to simultaneously reason about multiple fairness notions.
Example (Accuracy Parity (AP) (Zafar et al., 2017a)). ) A model he is fair for Accuracy Parity when the probability of being correct is independent of the sensitive attribute, that is, Vr E E
PP (ho(2)) = v r) (ho(z) = v).
Tt means that we need partition the space into K = ||| groups and, Vr E ES, we define F(r) as the fairness level of group (r)
(T, P (he (z) # y) P (1)(2) * y = r)
(PP (s r) 1) (ho(x) # s= = r) +
Σ (ho())))
(r))(()
where the law of total probability was used to obtain the last equality. Thus, Accuracy Parity satisfies all our assumptions with c(1) = (s c((r) ps with r' + r, and C =0
It is worth noting that FairGrad applies to any fairness measure that respects the definition above, even when there is a large number of groups. However, the performance of FairGrad may degrade when there are only a few samples per group, as fairness estimations become unreliable. In this case, the risk is that the learned model is fair on the training set but does not generalize well to new examples. To circumvent some of these issues works such as Hobert-Johnson et al. (2018); Kearns et al. (2018) have extended fairness definitions to multi-group scttings and proposed mochanisms to optimize them. In this work, we focus on classical fairness definitions and keep the line of research to exlend Fairgrad to these allernative definitions for the future.
2.2 Related Work
Various in-processing methods have been proposed in the fair machine learning literature. Amongst them, many methods rely on formulating the problem either a constrained optimization, which is later relaxed to an unconstrained case or using re-weighting techniques where examples are dynamically re-weighed based on the fairness levels of the model (Caton & Haas 2020). In this sub-section, we will provide a brief overview of these methods and explain the similarities and differences between FairGrad and the corresponding ap- proaches. Additionally, we will also demonstrate how FairGrad can be seen as a solution that connects these two streams of work. For more details about very closely related works, please refer to Appendix A
Constrained Optimization The problem of fair machine learning can be seen as the following constrained
optimization problem (Cotter el al. 2019 Agarwal et al. 2018):
arg min PP (ha (x) + v)
s.t. Vk E [K], (T, h4) = 0.
(2)
This problem can then be reformulated as an unconstrained optimization problem using Lagrange multipli- ers. More specifically, with multipliers denoted by A1....,A , the unconstrained objective that should be minimized for he H and maximized for A AK is:
K
L (ho, A AR)= = (ho(z) + v) Σ (TTr)(
(3)
Several strategies may then be employed to find saddle point for this objective Agarwal et al. (2018) first relax the problem by searching for a distribution over the models rather than single optimal hypothesis.
These min-max formulations
new in the litcrature and already used in the 1940's Wald 1945. More
cently, Madry al. (0088) employed the formulation to make deep neural networks more robust. against adversarial attacks. Similarly, Ben- al. (2012) modeled uncertainty imput this formulation.
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Then, they alternate between using exponentiated gradient step to find A1,...,AA E R and procedure based on cost sensitive learning to find the next ho to add to their distribution. Similarly, Cotter et al. (2019) also search for a distribution over the models using an alternating approach based on Lagrange multipliers where they relax objective 3 by replacing the error rate with a loss term. To update the À multipliers,
unlike Agarwal et al. (2018) they use projected gradient descent based on the original fairness lerms. To search the next hg to add to their distribution of models they use a projected gradient descent update over a relaxed overall objective function where the fairness measures are replaced with smooth upper bounds.
In this work, we also use alternating approach based on objective 33 However, look for a single model rather than distribution of models. To this end, at each iteration, we update À using a projected gradient descent step similar to Cotter et al. (2019), that is using the original fairness measures. To solve for he, contrary to Cotter et al. (2019), we first show that Objective (3 with fixed X, may be rewritten as a weighted sum of group-wise error rates. This similar spirit to the cost-sensitive learning method of Agarwal et al.] (2018) but can be applicd beyond simple binary classification. We then follow Cotter et al.) (2019) and replace in our new objective the error rate terms with loss function, albeit not necessarily an upper bound, to obtain meaningful gradient directions.
Re-weighting Another way lo learn fair models is to use a re-weighting approach where each example æ is associated with a weight We E R so that minimizing the following objective for he outputs a fair model:
W (ha) = E (w>IIhec))+)))
The underlying idea for the methods which posit the problem as above is to propose cost function that outputs weights for each example. On the one hand, the weights can be determined in a pro-processing step (Kamiran& & Calders 2012), based on the statistics of the data under consideration. On the other hand, the weights may evolve with hep that is they are dynamically updated each time the model changes during the training process (Roh et al. 2020).
In this work, to find hg, we also use a dynamic re-weighting approach where the weights change at each iteration. To choose the weights, we initially give the same importance to cach example. Then, we increase the weights of vintavan examples and decrease the weights of advantaged examples proportionally to the fairness level of the current model for their group. An important feature of our approach, unlike other weightthting approaches, is that we do not constrain ourselves to positive weights but rather allow the use of negative weights. Indeed, we show in Lemma I that the latter are somctimes necessary to learn fair models.
To summarize, we first frame the task as a const rained optimization problem, similar to Cotter et al. (2019) and AAarwal et al. (0018). We then propose an alternating approach, where we update À at each iteration using projected gradient descent step similar to Cotter et al. (2019). However, in order to learn the model he; we show that Objective (3), with fixed X, can be rewritten as weighted sum of group-wise error rates. This step can be interpreted as an instance of dynamic reieeigh where the weights change at each iteration. Thus our method can be seen as a connection between constrained optimization and re-weighting,
3 FairGrad
In the previous section, we argued that FairGrad is connected to both constrainod optimization and re- weighting approaches. this section, we provide details on our method and we present it starting from the constrained optimization point of view as we believe it makes it casier to understand how the weights are selected and updated. We begin by discussing FairGrad for exact fairness and then extend it to e-fainness.
3.1 FairGrad for Exact Fairness
To solve the fairness problem described in equation 3 we propose to use an alternating approach where the hypothesis and the multipliers are updated one after the other We begin by describing method to update the multipliers and then the model.
It is worth noting that, here, we do have formal duality guarantees and that the problem is not even guaranteed to have fair solution. Nevertheless, the approach seems to work well in practice as can seen in the experiments.
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Updating the Multipliers. To update A1...., AK. we will use standard gradient ascent procedure. Hence, given that the gradient of Problem 3 18
(T(T,ho)
Fx (T))o))
we have the following update rule Vk E [K]: x
X 7y (T,
where 77 is rate that controls the importance of each update. In the experiments, we use constant rate of 0.01 initial tests showed that it is good rule of thumb when the data is properly standardized.
Updating the Model. To update the parameters 0 E Rº of the model ha, we use a standard gradient descent. However, first, we notice that, given our fairness definition, Equation 3
can be written as
K
K
L (ha,>1, ,
Ax)
P(al() + U(Th)
+ ((T))
Σ A #=1
+ AcC
(4)
k=1
k=1
where E Oa is independent of he by definition, Hence, at iteration t, the update rule becomes
8 8T 110 Σ
P Σ
Cl
VoP
TT)
k=1
where no is the usual learning rate that controls the importance of each parameter update. Here, we obtain our group specific weights Vx, wi (T.) + Ct Ax that depend on the current fairness level of
the model through A1...., AK, the relative size of each group through P (Tk), and the fairness notion under consideration through the constants C. The exact values of these constants Appendix
given in Section 2.1 and B for various group fairness notions. Overall, they are such that, at each iteration, the weights of the advantaged groups are reduced and the weights of the dasadvantaged groups are increased.
The main limitation of the above update rule is that one needs to compute the gradient of 0-1-sossss since VoP (ho(x) # T)Tx) = 1 (egyETT (oI(hu<)))))) Unfortumately, this usually does not provide meaningful optimization directions. "To address this issue, we follow the usual trend machine learning and replace the 0- 1-Iss with one of its continuous and differentiable surrogates that provides meaningful gradients. For instance, in our experiments, use the cross entropy loss.
3.2 Computational Overhead of FairGrad.
We summarize our approach in Algorithm [ where have used italic font to highlight the steps inherent to FairGrad that do not appear in classic gradient descent. We consider batch gradient descent rather than full gradient descent as it is popular scheme. We empirically investigate the impact of the batch size in Section 4.7 The main difference is Step (in italic font), that is the computation of the group-wise fairness levels. However, these be cheaply obtained from the predictions of (( the current batch which are always available since they also needed to compute the gradient. Hence, the computational overhead of FairGrad is very limited.
3.3 Importance of Negative Weights.
A key property of FairGrad is that we allow the use of negative weights, that is (Tx) - Σk=
K== C Ap may
become negative, while existing methods (Roh et al. 2020 Iosifidis & Ntoutsi 2019 Jiang & Nachum 2020)
restrict themselves to positive weights. In this section, we show that these negative weights are important they are sometimes necessary to learn fair models. Hence, in the next lemma, we provide sufficient conditions that negative weights mandatory if one wants to enforce Accuracy Parity.

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Algorithm PairGrad for Exact Fairness
Input: Groups Ti...., Tk, Functions Fi,..., Fk, Function class H of models he with parameters R Learning rates 7e, and Iterator iter that returns batches of examples. Output: A fair model hg.
1: Initialize the group specific weights and the model.
for B in iter do
Compute the predictions of the current model on the batch B.
4: Compute the group-wise losses using the predictions.
5: Compute the current fairness level using the predictions and update the group-wise weights.
6: Compute the overall weighted loss using the group-wise weighls.
7: Compute the gradients based on the loss and update the model.
end for
return the trained model hig
Lemma 1 (Negative weights are necessary.). Let the fairness notion be Accuracy Parity (Example 1 h be the most accurate and fair model. Then using negative weights is necessary as long as
Let
min max Po (h)( + u(Tk) < (h)((a) h bg unfair
+ tv)
Te
Proof. The proof' is provided in Appendix
C
The previous condition can sometimes be verified in practice. As motivating example, assume a binary setting with only two sensitive groups Ti and T-1. Let h be the model minimizing IP (ba(x) + y|T-1) and assume that P (hm'(x) + y) PP (h (x)) + v|T-1), that is group T_1 is disadvantagod for accuracy parity. Given h the most accurate and fair model, we have
min max P (ho( + TTT) = (lhg'1) # vTT < (1)(()) +
baER Te hyuunaiir
as otherwise we would have contradiction since the fair model would also be the most accurate model for group T-1 since P (hg(#) # y) =P P (hz(2) # v/T-1) by definition of Accuracy Parity. In other words, a dataset where the most accurate model for a given group still disadvantages requires negative weights. This might be connected to the notion of leveling down (Zietlow et al., 2022 Mittelstadt al. 2023), where fairness can only be achieved by harming all the groups or bringing advantaged groups closer to disadvantaged groups by artifact of strictly egalitarian fairness measures. Investigating this negative
harming them. It is generally effect is an important rescarch dircction that goes beyond the scope of this paper. Nevertheless, potential solution to mitigale it is to use her kind of fairness definitions. As a first step in this direction, in the next section we extend FairGrad to e-fairness where strict equality relaxed.
3.4 FairGrad for e-fairness
In the previous section, we considered exact fairness and we showed that this could be achieved by using a re-weighting approach. Here, we extend this procedure to e-fairness where the fairness constraints are relaxed and controlled amount of violations is allowed. Usually, is a user defined parameter but it can also be set by the law, as is the case with the 80% rule in the US (Biddle) 2006). The main difference with exact fairness is that cach equality constraint in Problem (2)
is replaced with two incqualitics of the form
Yk E [K], PT ho) S
Vk: E [K], (M(T ho) -E.
The main consequence that we need maintain twice as many Lagrange multipliers and that the group- wise weights are slightly different. Since the two procedures are similar, we omit the details here but provide them in Appendix [2 for the sake of completeness.

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4 Experiments
In this section, present several experiments that demonstrate the competitiveness of FairGrad as procedure to learn lair models for classification. We begin by presenting results over standard fairness datasets and a Natural language Processing dataset in Section 4.4 We then study the behaviour of the e-fairness variant of FairGrad in Scction 4.5 Next, we showcase the fino-tuning ability of FairGrad on a Computer Vision dataset in Section 4.6 Finally, we investigate the impact of batch size on the learned model in Section and present results related to the computational overhead incurred by FairGrad in Section 4.8
4.1 Datasets
In the main paper, consider different datasets and postpone the results another 6 datasets to Appendix E.3 as they follow similar trends. We also postpone the detailed descriptions of these datasets as well as the pre-processing steps to Appendix F.2
We consider commonly used fairness datasets, namely Adult Income (Kohavi 1996) and CelebA LLu et al.) 2015). Both are binary classification datasets with binary sensitive attributes (gender). We also consider variant of the Adult Income dataset where we add a second binary sensitive attribute (race) to obtain a dataset with 4 disjoint sensitive groups. For both datasets, use 20% of the data test set and the remaining 80% as a train set. We further divide the train set into two and keep 25% of the training examples as a validation set. For each repetition, we randomly shuffle the data before splitting it, and thus we have unique splits for each random scod. Lastly, we standardize each features independently by subtracting the mean and scaling to unit variance which were estimated on the training set.
To showcase the wide applicability of FairGrad, we consider the Twitter Sentiment [Blodgett et al. 2016) dataset from the Natural Language Processing community. It consists of 200k tweets with binary sensitive attribute (race) and binary sentiment score. We employ the same setup, splits, and the pre-processing proposed by Han et al. (2021) and Flazar & Goldberg (2018) and create bias in the dataset by changing the proportion of each subgroup (race-sentiment) in the training set. Following the footsteps of Elazar & Goldberg (2018) we encode the tweets using the DeepMoji (Felbo et al.) (017)) encoder with no fine-tuning, which has been pre-trained over millions of tweets to predict their emoji, thereby predicting the sentiment. We also employ the UTKFace ] İZhang al., (017)) from the Computer Vision community. It consists of 23, 708 images tagged with race, age, and gender with pre-defined splits.
4.2 Performance Measures
For fairness, we consider the four measures introduced in Section 2.1 and Appendix B namely Equalized Odds (EOdds), Equality of Opportunity (FOpp), Accuracy Parity (AP), and Demographic Parity (DP). For each specific fairness notion, we report the average absolute fairness level of the different groups over the test set, that is IK (T) (lower better). To assess the utility of the learned models, we use their accuracy levels over the test set, that is 1 n Em= Inde))= (higher better). All the results reported are averaged over 5 independent. runs and standard deviations are provided. Note that, in the main paper, we graphically report subset of the results over the aforementioned datasets. We provide detailed results in Appendix E.3 including the missing pictures well as complete tables with accuracy levels, fairness levels, and fairness level of the most well-off and worst-off groups for all the relevant methods.
4.3 Methods
We compare FairGrad a wide variety of baselines, namely:
Unconstrained, which is oblivious to any fairness measure and is trained using standard batch gradient descent method.
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Adversarial learning based method where employ adversarial mechanism Goodffllow et al,
2014h using gradient reversal layer (Ganin & Lempitsky 2015), similar to GRAD-Pred (Raff & Sylvester 2018), where an adversary, with an objective to predict the sensitive attribute, is added to the unconstrained model
Bi-level optimization based method implemented in the form of BiFair (Ozdayi et al. 2021
Re-weighting based methods in the form of FairBatch (Roh ct al., 2020). We also compare against a simpler baseline called Weighted ERM where each example is reweighed based on the size of the sensitive group the example belongs to in the beginning. Unlike FairBatch these weights are not. updated during training.
Constrained optimization based method as proposed by Cotter et al. (001 We refer to this method as Constraints in this article.
Reduction implements the exponentiated gradient based fair classification approach proposed by Agarwal et al. 2018
In all our experiments, we consider two different hypothesis classes. the one hand, we use linear models implemented in the form of neural networks with no hidden layers. On the other hand, we use a more complex, non-lincar architecture with three fully-connected hidden layers of respective sizes 128, 64, and 32. We use ReLU as our aclivation function with batch normalization and dropout. In both cases, we optimize the cross-entropy loss.
In several experiments, we only consider subscts of the basclincs due to the limitations of the mothods. For instance, Bifair was designed to handle binary labels and binary sensitive attributes and thus is not considered for the datasets with more than two sensitive groups or two labels. Furthermore, we implemented it using the authors code that is freely available online but does not include AP as a fairness measure, thus we do not report results related to this measure for BiFair. Similarly, we also implemented FairBatch from the authors code which does not support AP as fairness measure, thus we also exclude it from the comparison for this measure. For Constraints, we based our implementation on the publicly available authors library but were only able to reliably handle linear models and thus we do not consider this baseline for non-linear models. Finally, for Adversarial, we used our custom made implementation. However, it only applicable when learning non-linear models since requires at least one hidden layer to propagate its reversod gradient.
Apart from the common hyper-parameters such as dropout, several baselines come with their own set of ppper-parametess For instance, BiFair has the inner loop length, which controls the number of iterations in its inner loop, while Adversarial has the scaling, which re-weights the adversarial branch loss and the task loss. We provide details of common and approach specific hyper-parameters with their range in Appendix E.
With several parameers for each approach, selecting the best combination is often crucial to avoid undesirable behaviors such as over-fitting (Maheshwari et al.) 2022). In this paper, we opt for the following procedure. First, for each method, we consider all the X possible hyper parammee combinations and we run the training procedure for 50 epochs for each combination. Then, last 5 epochs, that is, for a given method, we have 5XX models and the goal is to select the best one among them. Since we have access to two performance measures, we can select either the most accurate model, the most fair, or trade-off between the two depending on the end goal. Here, we chose to focus on the third option and select the model with the lowest fairness score between certain accuracy intervals. More specifically, let a* be the highest validation accuracy among the 5X models. We choose the model with the lowest validation fairness score amongst all models with a validation accuracy in the interval [a" k; a*). In this work, we fix k to 0.03.
retain all the models returned by the
4.4 Results for Exact Fairness
We report the results over the Adult Income dataset using linear model, the Adult Income dataset with multiple groups with a non-linear model, and the Twitter sentiment dataset using both linear and nonlinear and respectively. In these figures, the best methods closer to the bottom right
models in Figures 2 g
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corner. If a method is closer to the bottom left corner, it has good fairness but reduced accuracy. Similarly, method closer to the top right corner has good accuracy but poor fairness.
The main take-away from these experiments that there is no fairness enforcing method that is consistently better than the others in terms both accuracy and fairness. All of them have strengths, that is datasets and fairness measures where they obtain good results, and weaknesses, that datasets and fairness measures for which they are sub-optimal. FairBatch induces better accuracy than the other approaches over Adult with linear model and EOdds and only pays small price in fairness. However, it is significantly worse in terms of fairness over the Adult Multigroup dataset with non-lingg non-linear model. Similarly, BiFair is sub-optimal on Adult with EOpp, while being comparable to the other approaches on the Twitter Sentiment dataset. We observed similar trends on the other datasets, available in Appendix E.3 with different methods coming out on top for different datasets and fairness measures.
Interestingly, FairGrad generally outperforms other approaches in terms of fairness, albeit with slight loss in accuracy. These observations are even more amplified in the Accuracy Parily and Equalized Odds settings. Moreover, it is generally more robust and tends to show a lower standard deviation in accuracy and fairness than the other approaches. Even in terms of accuracy, the largest difference is over the Crime dataset, where the difference between FairGrad and Unconstrained is 0.04. However, in most cases, the difference is within 0.02. In terms of the multi-group setup, we find similar observations, that is FairGrad outperforms other approaches in fairness, albeit with a drop in accuracy. In fact, for Equality of Opportunity FairGrad almost outperforms approaches in terms of fairness and accuracy. Overall, FairGrad performs reasonably well all the settings we considered with no obvious weaknesses, that is no datasets with the lowest accuracy and fairness compared to the baselines.
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Figure 2: Results for the Adult dataset using Linear Models.
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Figure 3: Results for the Adult Multigroup datasct using Non Linear models.
4.5 Accuracy Fairness Trade-off
In this second set of experiments, we demonstrate the capability of FairGrad to support approxinate fairness (see Section 3.4h. In Figure we show the performance, accuracy-fairne fairness pairs, of several models learned
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FairGrad
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Figure 4: Results for the Twitter Sentiment dataset for Lincar and Non Linear Models.
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Figure 5: Results for CelebA using Linear models. The Unconstrained Linear model achieves a test accuracy of 0.8532 with fairness level of 0.0499 for EOdds, 0.0204 for AP, and 0.0387 for EOpp.
on the CelebA dataset by varying the fairness level parameter e. These results suggest that FairGrad respects the constraints well. Indeed, the average absolute fairness level (across all the groups, Section 4.2) achieved by FairGrad is either the same less than the given threshold. It is worth mentioning that FairGrad is designed to enforce e-fairness for each constraint individually which is slightly different from the summarized quantity displayed here. Finally, as the fairness constraint is relaxed, the accuracy of the model increases, reaching the same performance as Unconstrained when the fairness level the latter is below c.
4.6 FairGrad as Fine- Mine-Tnning Procedure
While FairGrad has primarily been designed to learn fair classifiers from scratch, it can also be used to fine- tune an existing classifier to achieve better fairness. To showcase this, we fine-tune the ResNet18 (He el al. 2016) model, developed for image recognition, over the UTKFaae dataset (Zhang et al. 2017). consisting of human face images tagged with Gender, Age, and Race information. Following the same process as Roh al. (2020g, we use Race the sensitive attribute and consider two scenarios. Either we consider Demographic Parity as the fairness measure and use the gender (binary) the target label consider Equalized Odds
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Table 1:
Results for the UTKFaae dataset where a ResNet18 is fine tuned using different strategies.
8- Race
yeGeder
S Race
Age
Method
Accuracy
DP
Accuracy
EOdds
Unconstrained 0.8691 + 0.0075 0.0448 + 0.0066 0.6874 0.0080
0.0843 0.0089 0.0506 0.0059
FairGrad
0.8397 + 0.0085 0.0111 + 0.0064 0.6491
0.0082
Table 2:
Batch size effect on the CelebA dataset with Linear Models and EOdds the fairness measure.
Batch Size
8
16
32
64
128
256
512
1024
2048
Accuracy
0.8186
0.8234
0.8215
0.8268
0.8273
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0.0013
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and predict the age (multi-valued). The results are displayed in Table | In both settings, FairGrad learns
models that are more fair than an Unconstrained fine-tuning procedure, albeit at the expense of accuracy.
4.7 Impact of the Batch-size
In this section, we evaluate the impact of batch size on the fairness and accuracy level of the learned model. Indeed, at each iteration, in order to minimize the overhead associated with FairGrad (see Section 3.1 we update the weights using the fairness level of the model estimated solely on the current batch. When these batches are small, these estimates are unreliable and might lead the model astray. In Table 2 we present the performances of several linear models learned with different batch sizes on the CelebA dataset. Over this dataset, we observe that FairGrad consistently learns a fair model across all batch sizes and obtains reasonable accuracy since Unconstrained has an accuracy of 0.8532 for this problem. Nevertheloss, we still recommend the practitioners use a larger batch size whenever possible as we observe slight reduction in terms of fairness standard deviations.
4.8 Computational Overhead
In this last experiment, we evaluate the overhead of FairGrad, by reporting the wall clock time in seconds to train for an epoch with the Unconstrained approach and our method in various scttings.
We show the effect of model size by varying the number of hidden layers of the model over the Adult Income dataset, which consists of 45, 222 records. We used an Intel Xeon E5-2680 CPU train.
2016 fine tuned over the We consider a large convolutional neural network (ResNet18 (He et al. UTK- dataset consisting of 23, 708 images. We trained the model using a Tesla P100 GPU.
We experiment with a large transformer (bert-basc-uneased) @Devlin et al. 2019 fine tuned over the Twitter Sentiment Dataset consisting of 200k tweets. We trained using a Tesla P100 GPU.
We present results of the computation overhead of FairGrad in Table We find that the overhead is limited and should not be critical in most applications as it does not depend on the complexity of the model but, instead, on the number of examples and the batch size. Overall, these observations are in line with the arguments presented in Scction 3.2
5 Conclusion
In this paper, we proposed FairGrad, fairness aware gradient descent approach based on re-weighting scheme. We showed that it can be used to learn fair models for various group fairness definitions and is able
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Table 3: The computational overhead of FairGrad various settings. BS here refers to Batch Size, and the Unconstrained and FairGrad columns refers the average time in seconds taken by these approaches for an epoch, respectively. Delta refers the difference in time between these two approaches.
Setting
Parameters
BS
Unconstrained
FairGrad
Delta
Linear model Adult Dataset -CPU
106
512
0.277 + 0.031
0.307 + 0.01
0.03
2 layers -Adult Dataset -CPU
1762
512
0.315 + 0.036
0.316 + 0.029
0.01
5 layers -Adult Dataset -CPU
21346
512
0.370 + 0.042
0.394 + 0.025
0.02
10 layers -dult Dataset -CPU
39042
512
0,483 0.021
0.499 = 0.034
0.02
20 layers -ddult Datasct -CPU
80642
512
0.672
0.034
0.689 + 0.026
0.02
ResNet18 trained -UTKFace -GPU
11177538
64
31.173 = 0.085
31.588 0.055
0.42
Bert Twitter Sentiment -GPU
109505310
32
2246.342 + 3.20
2294.382 + 4.01
48.04
to handle multiclass problems as well as settings where there is multiple sensitive groups. We empirically showed the competitiveness of our approach against several baselines on standard fairness datasets and on a Natural Language Processing task. We also showed that it can be used to fine-tune an existing model on Computer Vision task. Finally, since based on gradient descent. and has a small overhead, we believe that FairGrad could be used for wide range of applications, even beyond classification.
Limitations and Societal Impact
While appealing, FairGrad also has limitations. It implicitly assumes that a set of weights that would lead to a fair model exists but this might be difficult to verify in practice. Thus, even if in our experiments FairGrad seems to behave quite well, practitioner using this approach should not trust it blindly. It remains important to always check the actual fairness level of the learned model. On the other hand, we believe that, due to its simplicity and its versatility, FairGrad could be easily deployed in various practical contexts and, thus, could contribute to the dissemination of fair models.
Acknowledgements
This work has been supported by the Région Hauts de France (Projet STaRS Equité en apprentissage décentralisé respectueux de vie privée) and Agence Nationale de la Recherche under grant number ANR- 19-CE23-0022. The authors would also like to thank Michael Lohaus and anonymous reviewers for helpful discussions and feedbacks.
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Appendix
In this appendix, we provide details that were omitted in the main paper. First, in Section we review we show that several well known group fairness
several works closely related to ours. Then, in Section B
measures are compatible with FairGrad. Section the update rules for FairGrad with e-fairness. Finally, in Section we provide additional experiments.
C we prove Lemma [] Next, in Section D we derive
A Related Work
The fairness literature is extensive and we refer the interested reader to recent surveys (Caton & Haas 2020: Mchrabi et al.) 2021) to get an overview of the subject. Here, we focus recent works that are more closcly related to our approach.
BiFair (Ozdayi et al., 2021 This paper proposes a bilevel optimization scheme for fairness. The idea is to use an outer optimization scheme that learns weights for each example so that the trade-off between fairness and accuracy is as favorable as possible while an inner optimization scheme lcarns a model that is as accurate as possible. One limitation of this approach is that it does not directly optimize the fairness level of the model but rather relaxation that does not provide any guarantees on the goodness of the learned predictor. Furthermore, it limited binary classification with a binary sensitive attribute. In this paper, we also learn weights for the examples in iterative way. However, we different update rule. Furthermore, we focus on exact fairness definitions rather than relaxations and our objective is to learn accurate models with given levels of fairness rather than a trade-off between the two. Finally, our approach is not limited to the binary sctting.
FairBatch (Roh et al., 2020). This paper proposes a batch gradient descent approach to learn fair models. More precisely, the idea is to draw a batch of examples from a skewed distribution that favors the disadvantaged groups by oversampling them. In this paper, we propose to use re- reiieighting approach which could also be interpreted as altering the distribution of the examples based on their fairness level if all the weights were positive. However, we allow the use of negative weights, and we prove that they are sometimes necessary to achieve fairness. Furthermore, employ different update rule for the weights.
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AdaFair (Iosifidis & Ntoutsi, 2019). This paper proposes boosting based framework to learn fair models. The underlying idea is to modify the weights of the examples depending on both the performances of the current strong classifier and the group memberships. Hence, examples that belong to the disadvantaged group and are incorrectly classified receive higher weights than the examples that belong to the advantaged group and are correctly classified. In this paper, we use a similar high level idea but we use different weights that do not depend on the accuracy of the model but solely on its fairness. Furthermore, rather than a boosting based approach, we consider problems that can be solved using gradient descent. Finally, while AdaFair only focuses Equalized Odds, we show that our approach works with several fairness notions.
Identifying and Correcting Label Bias in Machine Learning (Jiang & Nachum 2020). This paper tackles the fairness problem by assuming that the observed labels are biased compared to the true labels. The goal is then to learn a model with respect to the true labels using only the observed labels. To this end, it proposes to use an iterative weightght procedure where positive example-wise weights and the model are alternatively updated. In this paper, we also propose a re-weighting approach. However, we use different weights that are not necessarily positive. Furthermore, our approach not limited to binary labels and can handle multiclass problems.
B Reformulation of Various Group Fairness Notion
In this section, we present several group fairness notions which respect our fairness definition presented in Section 2.1
Example 2 (Equalized Odds (EOdds) (Hardt et al., 2016)) ). A model hg is fair for Equalized Odds when the probability of predicting the correct label is independent of the sensitive attribute, that is, Vi e y, Vr S
P (ha(z)) = =||s r,y = P (h)(x) = llv 1).
It mcans that we need to partition the space into = | x S groups and, VI E V. E S, we define P(n.)) a8
Purr) (o)) = (h)(2)) *lly =1 - P (he(a) +1|8=r,y
Σ
( = r'ly P (ho # =
(L.r)
- -11 PP (8= rly = 1)) (ho (x) # 1 =1)
where the law of total probability was used to obtain the last equation. Thus, Equalized Odds satisfies all Te (s r' 1), c(l' with r' * and for given subset ) y of labels called the desirable outcomes, that is, vI )", Vr E
our assumptions with c(lrr) P (s rly 1) 1, cllrr)
(I.r)
(1.r)
I' + 1, and C(.r) (1.))= 0.
Example 3 (Equality of Opportunity (EOpp) (Hardt et al.) 2016) ) A model he fair for Equality of Opportunity when the probability of predicting the correct label is independent of the sensitive attribute
P (2) 1| r. = P (h)(x) 11y 1)
It means that we need to partition the space into K = y x S groups and, VI E y, Vr E S, we define Pars as
{
P (ho(z) (|l=))
(T,
- (ho()))) = ((y=1) V(,r) x V(,r) EVx
which can then be rewritten in the correct form in the same way as Equalized Odds, the only difference being that Gi.,= = 0, v(I,r) E x {\) S.
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Example 4 (Demographic Parity (DP) (Calders et al., 2009)): ). A model hy is fair for Demographic Parity when the probability of predicting a binary label is independent of the sensitive attribute, that is. VI E y/ &
(ho((z) 1| r) PP (ho(2)) = 1)-
It means that we need to partition the space into K = > S groups and, VI y, Vr S, we define P(a) as
Pun (T, ho) = P (h) + - (ho(()) # r)
(P = 1,,= r)
- P (u = 1| 8 = r) Te (h =y|==
r') PP P -y=1,8 -,
Σ
( 1)
(r))(())))
= 1, -
( ( 11)
Σ Pyy=
P (bll
(Lr)(())))
P (y - 1) - P (u =1 r)
where the law of total probability was used to obtain the last equation. Thus, Demographic Parity satisfies
all our assumptions with c1,,) = P (y = 1,8 s= r) - P (y = 1|s = r), gilrrr
(1.r) = P (v = 1,s r') with r' * r, cllrr) = P (y= [y=[[s==] P (y I, 1,8=1) = r), c/ir' - (y = -
- r') with r' + r, and C(.,.) = PP (y = 1)
P (yy=[s=-r)
C Proof of Lemma 1
Lemma (Negative weights are necessary.). Assume that the fairness notion under consideration is Accuracy Pority, Let hig be the most accurate and fair model. Then using negative weights is necessary as long as
min max (ho(w)) 4 v(Th)) < (hū(a)) v)
h Te by Aguffair
Proof. To prove this Lemma, one first need to notice that, for Accuracy Parity, since E (Tx) = 11 we have that
K
(P(h)
K Σ
cK (P
k==1 (Tx.) = O.
This implies that
K P (Tx) + Σ k=1
K
c
=1
This implies that, whatever our choice of X, the weights will always sun to one. In other words, since we also have that Σk_ AaCh = 0 by definition, for given hypothesis he, we have that
K P (he(z) # u(TT)
K
max A
(Tx)
+
Am
(5)
ER
max
P (ho(x) fy(Ta) wk
(6)
Wi.... w. ER 8. Σ, w+=1
k=1
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where, given .1,.... wk, the original values of lambda can be obtained by solving the linear system CA where
Cl
Ck
A1
W1 (Ti)
C=
W
C
CR
Ak
WK P (Tk)
which is guarantoed to have infinitcly many solutions since the rank of the matrix C is K - 1 and the rank of the augmented matrix (Clw) is also K - 1. Here we are using the fact that PP (7k) + 0, vk since all the groups have to be represented to be taken into account.
We will now assume that all the weights are positive, that is wh 2 O, vk. Then, the best strategy to solve Problem (6) to put all the weight on the worst off group k, that set we = and wa = 0, vk' # implies that
Σ P (ho(x) # TT) wE max (ho(x) # v(Ta).
max MOK ER
s.t. wx=1
Furthermore, notice that, for fair models with respect to Accuracy Parity, we have that P (ho(rz) # yITT) P (hh v), vk. Thus, if it holds that
min max (ho + 3TTh) < P (hō(2) # hoER 7s hgunfair
where hy is the most accurate and fair model, then the optimal solution of Problem (3) in the main paper will be unfair. It implies that, in this case, using positive weights is not sufficient and negative weights are necessary.
D FairGrad for e-fairness
To derive FairGrad for e-fairness we first. consider the following standard optimization problem
arg miniP (2o + v)
hoE
s.t. Vk E [K], (T, S Vk E [K], (T, hq) 2 -E.
We, once again, use standard multipliers approach to obtain the following unconstrained formulation:
-) L (ha, Ay...., Ax, .x.,,,,..,6 = (ha(v) # #v) + A (Fa(T, -c) -c) - 8e (Fa(T,ho) +c)
6x)
(7)
1=1
where Ai,..., Ag and 81 again, to solve this problem, will alternating approach where the hypothesis and the multipliers are updated one after the other.
. ,g are the multipliers that belong to R+, that the set of positive reals. Once
Updating the Multipliers. To update the values A Ak, we will use standard gradient ascent procedure. Hence, noting that the gradient of the previous formulation is
FT hha -
V Ag L (he, .....,, AR. (x..,....6kk))
(x
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- (T((Thh)) -
V V 8g C (ha, Ak, (x))
(T, (T,ho)
we have the following update rule Vk E [K]
- max (0. - (Pa 1/0) -e))
%
max (0.8% - n (Fa (T,TT,, hb) + 2))
where 17 is a fairness rate that controls the importance of each weight update.
Updating the Model. To update the paramcters 0 E RD of the model has we proceed as before, using a gradient descent approach. However, first, we notice that given the fairness notions that we consider, is equivalent to
Equation
C (ho, A λK. .1,...,00r .k) Σ P (ho ulTk)
(Tx)
Σ
ch (Ax 5Md)
(8)
K (Ax + 6x) c +
K
Σ k=1
(Ax
Since the additional terms in the optimization problem do not depend on ha, the main difference between exact and e-fairness is the nature of the weights. More precisely, at iteration the update rule becomes
0-+1
0T =
P (T&)
Σ
ch Ow 5x)) VoP (ho(2) + Tp))
T7
where no a learning rate. Once again, we obtain a simple re-weighting scheme where the weights depend on the current fairness level of the model through At.... :y....;Ar Ak and , the relative size each group through P (Tk), and the fairness notion through the constants C.
E Extended Experiments
In this section, provide additional details related to the basclines and the hyper-parameters tuning procedure. We then provide descriptions of the datasets and finally the results.
E.1 Baselines
Adversarial: One of the common ways of removing sensitive information from the model's repre- sentation is via adversarial learning. Broadly, it consists of three components, namely an encoder, a task classifier, and an adversary. On the one hand, the objective the adversary to predict sensitive information from the encoder. On the other hand, the encoder aims to create represen- tations that are useful for the downstream task (task classifier) and, at the same time, fool the adversary. The adversary is generally connected to the encoder via gradient reversal layer (Ganin) & Lempitsky 2015) which acts like an identity function during the forward pass and scales the loss with a parameter -À during the backward pass. In our sctting, the encoder is Multi-Layer Perceptron with two hidden layers of size 64 and 128 respectively, and the lask classifier is another Multi-Layer Perceptron with a single hidden layer size 32. The adversary is the same as the main task classifier. We use a ReLU as the activation function with the dropout set 0.2 and employ batch normalization with default PyTorch parameters. As part the pyper-parameter tuning, we did grid search over X, varying it between 0.1 to 3.0 with interval of 0.2.
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BiFair (Ozdayi et al., 2021 ) For this baseline, we fix the weight parameter to be of length 8 suggested in the code released by the authors In this fixed setting, we perform a grid search over the following uyper-parameters:
Batch Size: 128, 256, 512
Weight Docay: 0.0, 0.001
Fairness Loss Weight: 0.5, 1, 2, 4
Inner Loop Length: 5, 25, 50
Constraints: We use the implementation available in the TensorFlow Constrained Optimization library with default param-ear
FairBatch: We use the implementation publicly released by the authors
Weighted ERM: We reweigh each example in the dataset based on inverse of the proportion of the sensitive group it belongs to.
Reduction: We use the implementation available in the Fairleart with default hyper- aaper-parameters.
In our initial experiments, we varied the batch size, and learning rates for both Constraints and FairBatch. However, we found that the defanlt syper-parameters as specified by the authors result the best per- formances. In the spirit of being comparable in terms of hyper-parameter search budget, we also fix all parrmeeeseeseer of FairGrad, apart from the batch size and weight decay. We experiment with two differ- ent batch sizes namely, 64 or 512 for the standard fairness dataset. Similarly, we also experiment with three weight decay values namely, 0.0, 0.001 and 0.01. Note that also vary weight decay and batch sizes for FairBatch, Adversarial, Unconstrained, and BiFair.
For all our experiments, apart from BiFair, we use Batch Gradient Descent as the optimizer with a learning rate of 0.1 and a gradient clipping of 0.05 to avoid exploding gradients. For BiFair, we employ the Adam optimizer as suggested by the authors with learning rate of 0.001. For FairGrad, FairBatch and Uncon- combinations. For BiFair, we considered 72 such combinations,
strained, we considered while for Adversarial, there 90 combinations.
E.2 Datasets
Here, we provide additional details on the datasets used in our experiments. We begin by describing the standard fairness datasets for which we follow the pre-processing procedure described in Lohaus et al. (2020).
Adult The dataset Kohavi 1996 is composed of 45222 instances, with 14 features each describing several attributes of a person. The objective is to predict the income of a person (below or above 50k) while remaining fair with respect to gender (binary in this case). Following the pre-processing step of Wu et al. (2019). only 9 features were used for training.
Celeb 10 The dataset LLu et al. 2015) consists of 202, 599 images, along with 40 binary attributes associated with each image. We use 38 of these as features while keeping gender as the sensitive attribute and "Smiling" as the class label.
Dutch The dataset (Zliobaite et al. 2011 is composed of 60, 420 instances with each instance described by 12 features. We predict "Low Income" or "High Income" as dictated by the occupation as the main classification task and gender the sensitive attribute.
hhttps://githan com htwoiboooommmmmmii Hatf /ati/ BBiFair
hitps::/ higgggtt.bb..o. com google- rew.leeerseeeecceeeeeee tensorflow constrained optimization
hilpps: http://ttttbbbb////ee www.ooooooooooo////ooooooooooo// lhilaibba
hitpps: wawrrccarn.co
hitpps/
/ttps:///rchivveee /archivc..cc... ICS uCI edu http:/ccc/m/aaass////ld/dd/// /atp//asaassttllllll
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Compas The dataset (Larson et al. 2016 contains 6172 data points, where each data point has 53 features. The goal is to predict if the defendant will be arrested again within two years of the decision. The sensitive attribute is race, which has been merged into "White" and "Non White" categories.
Communitics and Crime 13 The dataset [Redmond & Baveja, 2002) is composed of 1994 instances with 128 features, of which 29 have been dropped. The objective is to predict the nunber of violent crimes in the community, with race being the sensitive attribute.
German Credit The dataset (Dua al. 2017) consists of 1000 instances, with each having 20 attributes. The objective to predict person's 's creditworthiness (binary), with gender being the sensitive attribute.
Gaussiar It is toy dataset with binary task label and binary sensitive attribute, introducod in Lohaus et al. (2020). It is constructed by drawing points from different Gaussian distributions. We follow the same mechanism as described in Lohaus et al. (2020). and sample 50000 data points for each class.
Adult Folktables This dataset (Ding et al. 2021 an updated version of the original Adult In- come dataset. We use California census data with gender as the sensitive attribute. There 195665 instances, with 9 features describing several attributes of person. We use the same preprocessing step recommended by the authors.
For all these datasets, we use a 20% of the data as a test set and 80% as a train set. We further divide the train set into two and keep 25% of the training examples as a validation set. For each repetition, we randomly shuffle the data before splitting it, and thus we had unique splits for each random seed. We use the following seeds: 10, 20, 30, 40, 50 for all our experiments. As last pre-processing step, we centered and scaled each feature independently by substracting the mean and dividing by the standard deviation both of which were estimated on the training set.
Twitter Sentiment Analysis 17 The dataset Blodgett et al. 2016) consists of 200k tweets with binary sensitive attribute (race) and binary sentiment score. We follow the setup proposed by Han et al. (2021) and Elazar Goldberg (2018) and create bias in the dataset by changing the proportion of each subgroup (raco-sentiment) in the training set. With two sentiment classes being happy and sad, and two race classes being AAE and SAE, the training data consists of 40% AAE-happy, 10% AAE-sad, 10% SAE-happy, and 40% SAE-sad. The test set remains balanced. The tweets are encoded using the Deep.ooji (Felbo et al,, 2017 encoder with no fine-tuning which has been pre-trained over millions of tweets to predict their emoji, thereby predicting the sentiment. Note that the train-test splits are pre-defined and thus do not change based on the random seed of the repetition.
E.3 Detailed Results
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fadd crime
hittps:/ /trchhrc..ee.... sec..du//ai m/l datassts /tatittttstStttogogo Statlog 728Cerman Credit Data
htttps://guii http://gihhub..o. com. michaus hStpas///aaaraaaaaaaaaaaaaaa master cxwmples///////cccceeeceee get synthotic data.py
hhttps:/ httpps///gituuu..com///// toltaables
wtw.aaggla.....mmsse....... willerAAAF)
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FairGrad
FairGrad
FairGrad
0.08
Unconstrained
FairBatch
Equal Opportunity
0.05
FairBatch
Accuracy Parity
8.06
0.07
Weighted
Reduction
Equal Odds
0.06
0.07
Weighted Erm
0.06
0.07
Weighted Emm
3.00
Constraints
0.05
Reduction
0.05
Reduction
0.00
0.01
Constraints
0.01
0.03
0.03
BiFair
0.03
BiFair
01012
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.67
0.71
0.75
0.80
0.84
0.88
0.67
0.71
0.75
0.80
0.84
0.88
0.67
0.71
0.75
0.80
0.84
0.88
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.09
FairGrad
FairGrad
FairGrad
0.08
Accuracy Parity
0.06
0.00
Unconstrained
Weighted Erm
Equal Odds
0.06
0.07
FairBatch
Adversarial
Unconstrained
Equal Opportunity
0.06
0.07
FairBatch
Adversorial
3.05
Reduction
0.05
Weighted Erm
0.05
Weighted Enm
0.04
0.04
0.04
Reduction
0.03
0.03
BiFair
0.03
Bifair
0.02
0.02
0.02
0.01
0.01
0.01
0.0g
0.00
0.00
0.67
0.71
0.75
0.80
0.84
0.88
0.67
0.71
0.75
0.80
0.84
0.88
0.67
0.71
0.75
0.80
0.84
0.88
Accuracy
Accuracy
Accuracy
(d) Non Linear
(e) Non Linear F.Odds
(f) Non Linear E.Op
Figure 6: Results for the Adult dataset with different fairness measures.
Table 4: Results for the Adult dataset with Lincar Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY 1
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.8456 + 0.0033
AP
0.0571 + 0.0022
0.077 1\t 0.0029
-0.0373 0.0017
Constant
0.751 + 0.0
AP
0.102 + 0.0
0.138 + 0.0
0.067 = 0.0
Weighted ERM
0.842 + 0.0016
AP
0.0581 + 0.0021
0.0783 0.0028
-0.0379 + 0.0014
Constrained
0.783 + 0.007
AP
0.005 + 0.003
0.007 + 0.005
0.001 + 0.002
Reduction
0.7064 = 0.0315
AP
0.0361 + 0.0158
0.0235 + 0.0103
-0.0487 = 0.0214
FairGrad
0.8124 = 0.005
AP
0.0097 + 0.0029
0.0131 0.004
-0.0063 = 0.0019
Unconstrained
0.846 + 0.0028
Eodds
0.0453 + 0.0039
0.048 + 0.0043
-0.0878 + 0.01
Constant
0.748 + 0.0
Eodds
0.0 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.8475 = 0.0024
Eodds
0.044 + 0.0043
0.0477 0.0031
-0.0837 = 0.0124
Constrained
0.805 1 0.004
Eodds
0.007 +: 0.005
0.019 1: 0.017
0.002 + 0.001
BiFair
0.793 0.009
Eodds
0.036 1: 0.008
0.085 1 0.027
-0.03 + 0.016
FairBatch
0.8437 + 0.0013
Eodds
0.0228 + 0.0071
0.0411 + 0.0105
-0.0245 + 0.0183
Reduction
0.7059 + 0.0277
Eodds
0.0542 + 0.0158
0.0711 0.0189
1055 0.022
FairGrad
0.8284 = 0.004
Eodds
0.0051 + 0.0021
0.0078 + 0.0068
-0.0078 + 0.0054
Unconstrained
0.8457 + 0.0028
Eopp
0.0263 + 0.0024
0.0157 + 0.0011 -0.0893 + 0.0083
Constant
0.754 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 0.0
Weighted ERM
0.8475 + 0.0024
Eopp
0,0246 1: 0.0036
0.0148 + 0.002
-0.0837 + 0.0124
Constrained
0.846 0.002
Eopp
0.011 0.004
0.039 1 0.012
0.0
BiFair
0.8 + 0.009
Eopp
0.031 + 0.024
0.019 + 0.014
-0.107 + 0.083
FairBatch
0.8457 + 0.0016
Eopp
0.0098 + 0.0068
0.0225 0.0174
-0.0166 + 0.0211
Reduction
0.8226 + 0.0149
Eopp
0.0341 + 0.0168
0.116 + 0.0575
-0.200 + 0.0098
FairGrad
0.8353 = 0.0106
Eopp
0.0053 + 0.006
0.0177 + 0.021
-0.0037 = 0.0033
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Table 5: Results for the Adult dataset with Non Linear Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.8438 0.0025
AP
0.0575 + 0.0025
0.0776 + 0.0033 -0.0375 + 0.0018
Constant
0.751 + 0.0
AP
0.102 = 0.0
0.138 + 0.0
0.067 = 0.0
Weighted ERM
0.8469 0.0035
AP
0.0564 + 0.003
0.0761 + 0.0038
-0.0368 = 0.0021
Adversarial
0.8364 0.0063
AP
0.0526 + 0.0017
0.0709 + 0.0025
-0.0343 + 0.0009
Reduction
0.7015 + 0.0225
AP
0.0681 + 0.0184
0.014 + 0.0122 -0.0917 + 0.0247
FairGrad
0.8054 0.0051
AP
0.0034 0.0033
0.0033 + 0.0031 -0.0036 = 0.0042
Unconstrained
0.8299 + 0.0142
Eodds
0.0448 + 0.0109
0.0404 + 0.0136 -0.0977 + 0.0422
Constant
0.748 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8285 0.0085
Eodds
0.0102 0.0025
0.0196 + 0.0102
-0.0099 = 0.0047
Adversarial
0.8202 + 0.0068
Eodds
0.0145 = 0.0052
0.0288 + 0.0177
-0.0153 = 0.0067
Bilair
0.823 1 0.017
Eodds
0.038 1 0.009
0.09 + 0.034
-0.038 0.015
FairBatch
0.8379 1: 0.0009
Eodds
0.02 + 0.0088
0.0327 + 0.0153
-0.0244 1 0.0218
Reduction
0.729 + 0.0252
Eodds
0.0636 0.0176
0.0673 + 0.0203
-0.115 = 0.0334
FairGrad
0.827 + 0.0071
Eodds
0.0118 + 0.0024
0.022 + 0.014
-0.0165 + 0.0135
Unconstrained
0.8382 +: 0.0076
Eopp
0.0242 +: 0.0031
0.0145 + 0.0017 -0.0822 +: 0.0108
Constant
0.754 + 0.0
Eopp
0.0 = 0.0
0.0 0.0
0.0 0.0
Weighted ERM
0.8293 + 0.0091
Eopp
0.0051 + 0.0033
0.0141 + 0.0137
-0.0062 = 0.0038
Adversarial
0.8324 1 0.0058
Eopp
0.007 + 0.0044
0.0139 1 0.0159
-0.0144 + 0.0133
BiFair
0.815 1 0.014
Eopp
0.03 = 0.015
0.019 0.009
-0.103 0.053
FairBatch
0.8415 + 0.0054
Eopp
0.0082 + 0.0073
0.0157 + 0.0121
-0.017 0.0271
Reduction
0.8343 0.0059
Eopp
0.0294 + 0.0164
0.0779 + 0.0662
-0.0396
+ 0.0455
FairGrad
0.8373 + 0.0043
Eopp
0.0053 + 0.0047
0.0099 + 0.0146 -0.0112 + 0.0127
25

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Published in Transactions on Machine Learning Research (08/2023)
0.10
Fair Grad
FaiGGrdd
0.10
FairGrad
0.00
Unconstrained
0.00
FairBatch
Equal Opportunity
0.09
FairBatch
Accuracy Parity
0.07
0.08
Weighted
Reduction
Equal Odds
0.08
0.07
Unconstrained
Weighted Erm
0.07
0.08
Uncanstrained
Weighted Emn
0.06
Constraints
0.06
Reduction
0.06
Reduction
0.035
0.05
Constraints
0.05
Constraints
0.04
0.04
BiFair
0.04
BiFair
0.03
0.03
0..03
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.76
0.85
0.67
0.76
0.80
0.89
0.71
0.80
0.85
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOOp
0.10
FairGrad
FairGrad
0.10
FairGrad
Accuracy Parity
0.07
0.08
(100
Adversarial
Weighted Erm
Uncomtrriined
Equal Odds
0.09
0.07
Unconstrained
Adversarial
Equal Opportunity
0.00
0.07
0.08
FairBatch
Unconstrained
Adversarial
0.06
Reduction
0.00
Weighted
0.06
Weighted
0.05
Reduction
Reduction
0.00
0.04
BiFair
0.01
BäFair
0.03
0.03
0.03
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.76
0.67
0.80
0.89
0.80
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
Non Linear EOdds
(f) Non Linear EOpp
Figure 7: Results for the CelebA dataset with different fairness measures.
Table 6: Results for the ColebA dataset with Lincar Models. All the results are averagod over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY 1
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.8532 + 0.0009
AP
0.0204 + 0.0022
0.017 1\t 0.0019
-0.0238 + 0.0025
Constant
0.516 + 0.0
AP
0,072 + 0.0
0.084 + 0.0
0.06 = 0.0
Weighted ERM
0.853 0.0008
AP
0.0193 + 0.0021
0.0161 0.0018
-0.0225 + 0.0023
Constrained
0.799 + 0.013
AP
0.01 + 0.001
0.012 + 0.002
0.009 + 0.001
Reduction
0.7734 = 0.011
AP
0.0242 + 0.006
0.0282 + 0.0071
-0.0201 = 0.005
FairGrad
0.835 + 0.0028
AP
0.0012 + 0.0009
0.0011 0.0007
-0.0014 - 0.0011
Unconstrained
0.8532 + 0.0009
Eodds
0.0499 + 0.0019
0.0538 + 0.0024
-0.1011 + 0.0033
Constant
0.518 = 0.0
Eodds
0.0 + 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.853 0.0009
Eodds
0.0504 + 0.0019
0.0532 0.0024
-0.1001 = 0.0032
Constrained
0.802 0.004
Eodds
0.006 0.001
0.01 + 0.003
0.002 +: 0.001
BiFair
0.845 0.007
Eodds
0.021 1: 0.005
0.02 1 0.003
-0.036 +: 0.009
FairBatch
0.8518 + 0.0009
Eodds
0.0226 + 0.0017
0.0218 + 0.0028
-0.0411 + 0.0053
Reduction
0.7268 + 0.011
Eodds
0.0312 + 0.0036
0.0628 0.0089
-0.0334 0.0047
FairGrad
0.8274 = 0.002
Eodds
0.0025 + 0.0009
0.0038 + 0.0018
-0.0046 + 0.0026
Unconstrained
0.8532 + 0.0009
Eopp
0.0387 + 0.0014
0.0538 + 0.0024
-0.1011 + 0.0033
Constant
0.518 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 0.0
Weighted ERM
0.853 + 0.0008
Eopp
0,0383 1: 0.0014
0.0531 + 0.0024
-0.0999 + 0.0032
Constrained
0.834 0.005
Eopp
0.002 0.001
0.005 + 0.002
0.0
BiFair
0.848 + 0.004
Eopp
0.014 + 0.006
0.02 + 0.009
-0.037 + 0.017
FairBatch
0.8498 + 0.001
Eopp
0.0102 + 0.0016
0.0142 0.0022
-0.0268 + 0.0042
Reduction
0.7358 + 0.0159
Eopp
0.0698 + 0.0118
0.1821 + 0.0313
-0.0968 + 0.0158
FairGrad
0.844 + 0.0022
Eopp
0.0013 + 0.0009
0.0025 = 0.0021
-0.0028 = 0.0018
26

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Published in Transactions on Machine Learning Research (08/2023)
Table 7: Results for the CelebA dataset with Non Linear Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.8587 0.0015
AP
0.0184 + 0.0014
0.0154 + 0.0012 -0.0215 + 0.0016
Constant
0.516 + 0.0
AP
0.072 + 0.0
0.084 + 0.0
0.06 = 0.0
Weighted ERM
0.8593 + 0.0018
AP
0.018 + 0.0017
0.015 0.0014
-0.021 = 0.0019
Adversarial
0.8588 0.0012
AP
0.0178 + 0.0014
0.0148 + 0.0012
-0.0208 + 0.0015
Reduction
0.7802 + 0.0142
AP
0.0436 + 0.0108
0.0508 + 0.0123
-0.0364 + 0.0092
FairGrad
0.8359 0.0033
AP
0.0023 0.0012
0.0025 + 0.0015 -0.0021 = 0.0009
Unconstrained
0.8583 0.0012
Eodds
0.0432 + 0.003
0.0475 + 0.0028 -0.0893 + 0.0049
Constant
0.518 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8589 0.0009
Eodds
0.0419 0.0021
0.0459 + 0.0025
-0.0864 = 0.0038
Adversarial
0.8567 + 0.0014
Eodds
0.0223 = 0.002
0.0272 + 0.0039
-0.0511 = 0.0073
Bilair
0.856 1: 0.004
Eodds
0.023 + 0.002
0.028 0.005
-0.052 0.009
FairBatch
0.8533 1: 0.0037
Eodds
0.0217 +: 0.0014
0.0197 + 0.0026
-0.0321 1: 0.005
Reduction
0.7021 0.0323
Eodds
0.0813 + 0.0253
0.1777 + 0.0426
-0.0946 = 0.0238
FairGrad
0.8304 + 0.0031
Eodds
0.0037 + 0.0017
0.0048 + 0.0018 -0.0055 + 0.0023
Unconstrained
0.8585 +: 0.0016
Eopp
0.0341 +: 0.002
0.0473 +: 0.003
-0.0889 +: 0.0052
Constant
0.518 + 0.0
Eopp
0.0 = 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.859 + 0.0009
Eopp
0.0331 + 0.0014
0.046 + 0.0023
-0.0866 = 0.0035
Adversarial
0.8557 +: 0.0019
Eopp
0.0161 + 0.002
0.0223 1 0.0029
-0.0419 1 0.0053
BiFair
0.854 + 0.004
Eopp
0.015 + 0.009
0.021 0.012
-0.039 0.022
FairBatch
0.8475 + 0.0043
Eopp
0.0051 + 0.0024
0.007 + 0.0033
-0.0131 + 0.0063
Reduction
0.765 + 0.0149
Eopp
0.0533 + 0.0124
0.1393 + 0.033
-0.0738 + 0.0167
FairGrad
0.8439 + 0.0063
Eopp
0.0009 + 0.0008
0.002 0.0022
-0.0016 + 0.0011
27

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Published in Transactions on Machine Learning Research (08/2023)
0:25
Fair Grad
FaiGGrad
FairGrad
Unconstrained
FairBatch
Equal Opportunity
FairBatch
Accuracy Parity
0.20
Weighted
Equal Odds
0.20
Unconstrained
0.20
Uncanstrained
Reduction
Weighted Erm
Weighted Emn
0.15
Constraints
0.15
Reduction
0.15
Reductiion
Constraints BiFair
Constraints BiFair
0.10
0.10
0.10
0.05
0.05
0.00
0.00
0.00
0.85
0.63
0.79
0.90
0.68
0.79
0.85
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOOp
0.25
FairGrad
0.25
FairGrad
0.25
FairGrad
Accuracy Parity
0.20
Adversarial
Uncomtrainned
Equal Odds
Unconstrained
Equal Opportunity
0.20
FairBatch
Unconstrained
Weighted Erm
Adversarial
Adversarial
0.15
Reduction
0.15
Weighted
0.15
Weighted
Reduction
Reduction
0.30
0.10
BiFair
0.10
BäFair
0.05
0.05
0.00
0.00
0.00
0.74 Accuracy
0.63
0.79
0.90
0.68
0.79
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear EOdds
(f) Non Linear EOpp
Figure 8: Results for the Crime dataset with different. fairness measures.
Table 8: Results for the Crime datasct with Lincar Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.8145 + 0.0136
AP
0.0329 + 0.0195
0.0258 0.0162
-0.0399 + 0.0229
Constant
0.734 + 0.0
AP
0.272 + 0.0
0.377 + 0.0
0.168 = 0.0
Weighted ERM
0.808 0.0246
AP
0.0361 + 0.0108
0.0284
0.0091
-0.0438 + 0.0129
Constrained
0.775 + 0.015
AP
0.025 + 0.019
0.031 + 0.025
0.019 + 0.014
Reduction
0.8521 = 0.0075
AP
0.055 0.0197
0.0426 + 0.0147
-0.0673 = 0.0253
FairGrad
0.814 + 0.0102
AP
0.0403 + 0.0181
0.0316 0.0147
-0.049 + 0.0218
Unconstrained
0.8035 + 0.0212
Eodds
0.2152 + 0.0215
0.1038 + 0.0231
-0.396 + 0.0433
Constant
0.677 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.8045 = 0.0271
Eodds
0.2086 + 0.0357
0.0974 0.0165
-0.3747 + 0.0679
Constrained
0.751 1 0.014
Eodds
0.036 0.012
0.088 1: 0.043
0.007 + 0.004
BiFair
0.76 1: 0.03
Eodds
0.082 0.048
0.048 1 0.03
-0.163 1: 0.092
FairBatch
0.8306 + 0.0237
Eodds
0.2015 + 0.035
0.1054 + 0.0333
-0.3704 + 0.067
Reduction
0.6842 + 0.0339
Eodds
0.0611 + 0.0281
0.0349 0.0111
-0.1291
0.047
FairGrad
0.7634 = 0.03
Eodds
0.0938 + 0.0144
0.0491 + 0.016
-0.1927 + 0.0362
Unconstrained
0.804 + 0.0215
Eopp
0.1215 + 0.0183
0.1009 + 0.0238
-0.3852 + 0.0549
Constant
0.697 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 0.0
Weighted ERM
0.8171 + 0.0213
Eopp
0.1209 1: 0.0154
0,0985 + 0.0106
-0.3851 + 0.0599
Constrained
0.762 0.021
Eopp
0.044 0.021
0.138 0.066
0.0
BiFair
0.806 + 0.01
Eopp
0.085 + 0.038
0.073 + 0.042
-0.268 + 0.112
FairBatch
0.8225 + 0.0252
Eopp
0.1126 + 0.0259
0.1002 0.0281
-0.3501 + 0.0821
Reduction
0.6747 + 0.0488
Eopp
0.0283 + 0.022
0.0113 + 0.0375
-0.7718 + 0.0829
FairGrad
0.7755 = 0.0233
Eopp
0.0609 + 0.0149
0.0507 + 0.0166
-0.193 + 0.0456
28

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Published in Transactions on Machine Learning Research (08/2023)
Table 9: Results for the Crime dataset with Non Linear Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY
1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0,8165 + 0.019
AP
0.0535 + 0.0199
0.0423 + 0.0155 -0.0648 + 0.0251
Constant
0.734 + 0.0
AP
0.272 + 0.0
0.377 + 0.0
0.168 0.0
Weighted ERM
0.8271 + 0.0114
AP
0.0483 + 0.0167
0.0382 + 0.0139
-0.0584 = 0.02
Adversarial
0.809 + 0.0175
AP
0.0592 + 0.0173
0.0464 + 0.0133
-0.0719 + 0.0223
Reduction
0.8501 + 0.0096
AP
0.0559 + 0.0215
0.0432 + 0.0166 -0.0685 + 0.0269
FairGrad
0.822 + 0.0203
AP
0.0434 0.0206
0.0341 + 0.0162 -0.0326 = 0.0252
Unconstrained
0.8115 + 0.014
Eodds
0.1635 + 0.0395
0.0854 + 0.014
-0.3326 + 0.0649
Constant
0.677 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8135 0.0137
Eodds
0.1739 0.0394
0.0861 + 0.0212
-0.3309 = 0.0778
Adversarial
0.791 + 0.007
Eodds
0.1464 + 0.0168
0.0797 = 0.0192
-0.3001 = 0.0296
Bilair
0.793 1 0.022
Eodds
0.161 +: 0.032
0.091 0.025
-0.339
0.048
FairBatch
0.8391 + 0.0195
Eodds
0.189 + 0.0368
0.1106 + 0.0313
-0.3828 1 0.0671
Reduction
0.7258 0.0267
Eodds
0.0743 0.0409
0.0553 + 0.014
-0.1556 = 0.0976
FairGrad
0.7734 + 0.0251
Eodds
0.0982 + 0.0513
0.0511 + 0.0179
-0.2016 + 0.0771
Unconstrained
0.817 + 0.0152
Eopp
0.1044 +: 0.0133
0.0856 + 0.0123
-0.3321 +: 0.0489
Constant
0.697 + 0.0
Eopp
0.0 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.8205 + 0.0184
Eopp
0.1159 + 0.0191
0.0955 + 0.019
-0.368 = 0.0642
Adversarial
0.795 1 0.0148
Eopp
0.0959 + 0.0153
0.0802 + 0.0227
-0.3036 1 0.042
BiFair
0.807 + 0.025
Eopp
0.11 + 0.031
0.091 + 0.031
-0.351 0.097
FairBatch
0.8411 + 0.0177
Eopp
0.1217 + 0.0277
0.1083 + 0.0311
-0.3784 + 0.0891
Reduction
0.6887 0.0271
Eopp
0.0282 + 0.0159
0.034 0.0281
-0.0788 + 0.0619
FairGrad
0.7799 + 0.0243
Eopp
0.0675 + 0.0179
0.0556 + 0.0147
-0.2143
+ 0.0592
29

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Published in Transactions on Machine Learning Research (08/2023)
0.00
0.00
FairGrad
0.09
Accuracy Parity
0.08
Unconstriined
0.08
FairBatch
Equal Opportunity
0.08
FairBatch
0.07
Weighted Erm
Equal Odds
0.07
Unconstrained
0.07
Unconstrained
Reduction
Weighted Erm
Weighted
0.06
0.06
0.06
0.00
0.0G
Reduction
0.05
+
Reduction
0.01
0.01
0.011
0.03
0.03
0.03
0.02
0.02
0.02
0.00
0.00
0.00
0.61
0.66
0.72
0.77
0.83
0.88
0.61
0.66
0.72
0.77
0.83
0.88
0.61
0.66
0.72
0.77
0.83
0.88
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.10
0.10
0.10
0.09
FairGrad
FairGrad
FairGrad
Accuracy Parity
0.08
0.07
Adversarial
Unconstrained
Equal Odds
0.07
Unconstrained
FairBatch
Equal Oppartunity
0.08
0.07
FairBatch
Unconstrained
0.06
Weighted
0.00
Adversarial
0.06
Adversarial
Reduction
Weighted Erm
Weighted
0.05
0.05
0.05
Reduction
+
Reduction
0.00
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.61
0.56
0.72
0.77
0.83
0.88
0.61
0.66
0.72
0.77
0.83
0.88
0.61
0.66
0.72
0.77
0.83
0.88
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear EOdds
(f) Non Linear EOpp
Figure 9: Results for the Adult with multiple groups dataset with different fairness measures.
Table 10:
Results for the Adult with multiple groups dataset with Linear Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY 1
MEASURE MEAN ABS. + MAXIMUM
MINIMUM
Unconstrained
0.8451 + 0.0042
AP
0.0559 + 0.0047
0.0985 + 0.0111
-0.042 + 0.003
Coustant
0.754 +: 0.0
AP
0.097 0.0
0.159 1 0.0
0.024 +: 0.0
Weighted ERM
0.8454 +: 0.0032
AP
0.0562 + 0.0042
0.0993
0.0117
-0.0426 1 0.0018
Reduction
0.6436 + 010178
AP
0.049 + 0.01
0.0493 + 0.017
-0.0661 + 0.0113
FairGrad
0.807 0.0022
AP
0.0148 + 0.0041
0.0256 0.0048
-0.0107
0.0045
Unconstrained
0.844 0.0011
Eodds
0.0558 + 0.0062
0.0578
0.0069
-0.1586 1 0.0621
Constant
0.75 + 0.0
Eodds
0.0 + 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8448 + 0.0038
Eodds
0.0586 + 0.0097
0.0567
0.0048
-0.1702 0.0776
FairBatch
0.8396 = 0.0034
Eodds
0.0308 + 0.0057
0.0565 + 0.0116
-0.0641 + 0.0234
Reduction
0.6932 = 0.0264
Eodds
0.0446 = 0.0048
0.0806
0.043 -0.0896 0.0278
FairGrad
0.8162 = 0.0052
Eodds
0.0197 + 0.0118
0.0373 + 0.0233
-0.0493 + 0.0403
Unconstrained
0.8431 + 0.002
Eopp
0.0391 +: 0.0052
0.0297 + 0.0131
-0.169 +: 0.0565
Constant
0.762
Eopp
0.0 + 0.0
-0.1767 + 0.0797 0.0
Weighted ERM
0.8443 + 0.0038
Eopp
0.0415 + 0.01
0.0316 + 0.0145
FairBatch
0.8392 + 0.004
Eopp
0.0219 + 0.0055
0.05 0.0133
-0.0749 + 0.0285
Reduction
0.7615 + 0.0357
Eopp
0.026 + 0.0189
0.0887 + 0.0378
-0.1115 + 0.0867
FairGrad
0.834 + 0.0044
Eopp
0.0201 + 0.0099
0.0442 + 0.0415
-0.0679 + 0.0808
30

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Published in Transactions on Machine Learning Research (08/2023)
Table 11: Results for the Adult with multiple groups dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively,
FAIRNESS
METHOD (NL) ACCURACY 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.8427 + 0.0041
AP
0.0546 + 0.0026
0.0966 + 0.0098
-0.0421 + 0.0022
Constant
0.754 + 0.0
AP
0.097 = 0.0
0.159 + 0.0
0.024 0.0
Weighted ERM
0.8408 0.0031
AP
0.0575 + 0.0035
0.101 0.0106
-0.0443 = 0.0026
Adversarial
0.8358 0.0043
AP
0.0527 + 0.0028
0.0889 + 0.0066
-0.0401 + 0.0022
Reduction
0.7025 0.0144
AP
0.0388 + 0.0066
0.054 + 0.0151
-0.0525 + 0.0099
FairGrad
0.7991 0.0036
AP
0.013 0.0051
0.0257 + 0.0138 -0.0125 = 0.0043
Unconstrained
0.8347 0.0129
Eodds
0.0523 + 0.0126
0.0495 + 0.0166 -0.1772 + 0.0512
Constant
0.75 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8199 0.002
Eodds
0.0287 0.0076
0.0274 + 0.0177 -0.1013 = 0.0543
Adversarial
0.8251 + 0.0064
Eodds
0.0223 = 0.0065
0.0451 + 0.0308
-0.0667 = 0.0559
FairBatch
0.8212 0.0103
Eodds
0.0806 0.0137
0.0522 1 0.0076
-0.2545 1 0.0525
Reduction
0.7649 1 0.0241
Eodds
0.0386 1 0.011
0.044 1: 0.02 -0.0954 1 0.0465
FairGrad
0.8128 0.0102
Eodds
0.0196 + 0.0061
0.0392 + 0.0176 -0.0443 = 0.0342
Unconstrained
0.8373 0.0123
Eopp
0.0331 0.008
0.0183 + 0.0045 -0.1587 +: 0.0643
Constant
0.762 + 0.0
Eopp
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8216 0.0081
Eopp
0.0245 = 0.008
0.0243 + 0.0196
-0.1016 = 0.0543
Adversarial
0.8343 + 0.0036
Eopp
0.0209 + 0.0093
0.0327 + 0.013
-0.0927 = 0.0589
FairBatch
0.821 1 0.0097
Eopp
0.067 + 0.0168
0.047 1 0.0113
-0.2484 1 0.0535
Reduction
0.8156 1 0.0204
Eopp
0.0259 +:2 0.0209
0.0472 + 0.0325
-0.0968 1 0.1117
FairGrad
0.8341 + 0.0053
Eopp
0.0176 + 0.0059
0.0302 + 0.0272 -0.0731
+ 0.0543
31

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Published in Transactions on Machine Learning Research (08/2023)
0.10
Grad
0.10
FaiGGrad
0.10
FairGrad
Unconstrained
FairBatch
Opportunity
FairBatch
Accuracy Parity
0108
Weighted
Equal Odds
0,08
0.08
Reduction
Weighted Erm
Weighted Emn
3.00
Constraints
0,06
Reduction
0.006
Reductioon
0.05
Constraints
Constraints
0.04
0.04
BiFair
0..04
BiFair
0.02
0.02
0.02
0.00g
0.00
0.00
0.60
0.69
0.50
0.64
0.69
0.73
0.55
0.64
0.69
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.10
FairGrad
0.10
FairGrad
0.10
FairGrad
Accuracy Parity
Unconstrained
FairBatch
Equal Opportunity
FairBatch
0108
Adversarial
Weighted Em
Equal Odds
0.08
Adversarial
Unconstrained
0.08
Adversorial
Unconstrained
11.06
Reduction
0.06
Weighted Erm
0.06
Weighted
0.05
Reduction
Reduction
0.00
0.01
BiFair
0.01
BiFair
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.00
0.00
0.00
0.50
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear EOdds
(I) Non Linear EOpp
Figure 10: Results for the Compas dataset with different fairness measures.
Table 12: Results for the Compas dataset with Lincar Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.6644 + 0.0137
AP
0.0091 + 0.0025
0.0076 0.0031
-0.0107 + 0.004
Constant
0.545 + 0.0
AP
0,066 + 0.0
0.085 + 0.0
0.047 + 0.0
Weighted ERM
0.6671 + 0.0169
AP
0.0088 + 0.004
0.0061 0.0028
-0.0115 + 0.0051
Constrained
0.65 + 0.012
AP
0.014 + 0.005
0.018 + 0.006
0.009 + 0.003
Reduction
0.6141 = 0.011
AP
0.0107 + 0.0064
0.009 = 0.006
-0.0124 = 0.0086
FairGrad
0.6708 = 0.0166
AP
0.0083 = 0.0068
0.0057 0.0048
-0.0108 = 0.0088
Unconstrained
0.6636 + 0.0101
Eodds
0.0827 + 0.0165
0.0758 + 0.0133
-0.1553 + 0.0259
Constant
0.527 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.6685 = 0.0073
Eodds
0.082 + 0.0137
0.0697 0.0115
-0.1618 + 0.0222
Constrained
0.564 + 0.014
Eodds
0.007 +: 0.004
0.014 1: 0.011
0.002 + 0.001
BiFair
0.672 0.021
Eodds
0.076 0.023
0.071 1 0.025
-0.15 + 0.039
FairBatch
0.6847 + 0.0175
Eodds
0.09 + 0.0094
0.0854 + 0.0149
-0.1727 + 0.0304
Reduction
0.5493 + 0.027
Eodds
0.029 0.0058
0.0268 0.0062
-0.0622 0.0219
FairGrad
0.6557 = 0.0075
Eodds
0.0593 + 0.0128
0.0524 + 0.0102
-0.1241 + 0.0202
Unconstrained
0.6609 + 0.0106
Eopp
0.052 + 0.0107
0.062 + 0.0145
-0.1461 + 0.0286
Constant
0.55 = 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.6695 + 0.0055
Eopp
0.0554 1: 0.0074
0.0659 + 0.0107
-0.1557 + 0.0194
Constrained
0.565 0.015
Eopp
0.004 0.003
0.011 0.009
BiFair
0.68 + 0.013
Eopp
0.054 + 0.016
0.064 + 0.022
-0.15 + 0.044
FairBatch
0.6865 + 0.0171
Eopp
0.0618 + 0.0134
0.0715 0.0173
-0.1755 + 0.0364
Reduction
0.5828 + 0.0457
Eopp
0.0252 + 0.0178
0.03 + 0.0216
-0.7707 + 0.0498
FairGrad
0.6565 = 0.0152
Eopp
0.0467 + 0.0046
0.0071 0.0554
-0.1313 + 0.0119
32

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Published in Transactions on Machine Learning Research (08/2023)
Table 13: Results for the Compas dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.6593 0.0192
AP
0.0119 + 0.0072
0.0095 + 0.004
-0.0144 + 0.0107
Constant
0.545 + 0.0
AP
0.066 + 0.0
0.085 + 0.0
0.047 0.0
Weighted ERM
0.6687 + 0.0138
AP
0.0127 + 0.0061
0.011 0.0034
-0.0145 = 0.0099
Adversarial
0.6583 0.0157
AP
0.0078 + 0.0051
0.0066 + 0.0041
-0.009 0.0069
Reduction
0.6287 + 0.0117
AP
0.0118 + 0.0024
0.0103 + 0.0062
-0.0134 + 0.0024
FairGrad
0.6672 0.0099
AP
0.0113 0.005
0.0095 + 0.0023 -0.0131 = 0.0082
Unconstrained
0.6562 0.0154
Eodds
0.0782 + 0.014
0.0715 + 0.0136 -0.1521 + 0.0277
Constant
0.527 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.6615 0.0175
Eodds
0.0789 0.0131
0.0726 + 0.0077
-0.1496 = 0.0313
Adversarial
0,6504 + 0.0157
Eodds
0.059 = 0.0138
0.0549 + 0.0107
-0.1294 = 0.0183
Bilair
0.661 1: 0.009
Eodds
0.013
0.068 0.018
-0.133 0.016
FairBatch
0.6792 1 0.0086
Eodds
0.071 + 0.0083
0.0663 1 0.0091
-0.1508 1 0.0304
Reduction
0.5631 0.0072
Eodds
0.0214 + 0.0112
0.024 + 0.0102
-0.0489 = 0.0363
FairGrad
0.6457 + 0.0088
Eodds
0.061 + 0.0075
0.0564 + 0-0065
-0.127 + 0.0081
Unconstrained
0.6552 + 0.0137
Eopp
0.0553 +: 0.0108
0.0659 + 0.015
-0.1552 +: 0.0281
Constant
0.55 + 0.0
Eopp
0.0 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.6604 + 0.0163
Eopp
0.0519 + 0.0111
0.0618 + 0.0148
-0.1458 = 0.0299
Adversarial
0.6494 + 0.0148
Eopp
0.0472 + 0.0072
0.0563 1: 0.0108
-0.1327 1: 0.0183
BiFair
0.669 + 0.01
Eopp
0.042 + 0.02
0.05 + 0.025
-0.117 0.055
FairBatch
0.6802 + 0.0114
Eopp
0.0536 + 0.0133
0.062 + 0.0167
-0.1526 + 0.0367
Reduction
0.5801 0.0258
Eopp
0.025 + 0.0119
0.0296 + 0.0145
-0.0702
+ 0.0333
FairGrad
0.6586 + 0.0118
Eopp
0.0476 + 0.0056
0.0563 + 0.0067
-0.1339
+ 0.0163
33

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Published in Transactions on Machine Learning Research (08/2023)
0.05
Grad
0.05
FairGrad
0.05
FairGrad
Accuracy Parity
0.04
Weighted
Uncomstriined
Equal Odds
01.44
Unconstrained
FairBatch
Equal Opportunity
0.04
FairBatch
Uncanstrained
Reduction
Weighted Erm
Weighted Emn
0.03
Constraints
0.03
Reduction
0.03
Reducction
Constraints BiFair
Constraints BäFair
0.02
0.02
0.01
0.01
0.00
0.00
0.00
0.72
0.81
0.64
0.72
0.77
0.85
0.64
0.68
0.72
0.77
0.81
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.05
FairGrad
FairGrad
FairGrad
Accuracy Parity
0.04
Adversarial
Uncomttrinned
0.04
Unconstrained
Equal Opportunity
0.01
FairBatch
Unconstrained
Equal Odds
Weighted Erm
Adversarial
Adversarial
0.03
Reduction
0.03
Weighted
0.03
Weighted
Reduction
Reduction
0.02
0.02
BiFair
0.02
BäFair
0.00
0.00
0.00
0.81
0.64
0.77
0.68
0.77
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear EOdds
(f) Non Linear EOpp
Figure 11: Results for the Dutch dataset with different fairness measures.
Table 14: Results for the Dutch dataset with Lincar Models. All the results are averagod over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1 MAXIMUM
MINIMUM
Unconstrained
0.8049 + 0.007
AP
0.0281 + 0.006
0.0281 0.006
-0.0282 + 0.0061
Constant
0.524 + 0.0
AP
0.151 + 0.0
0.152 + 0.0
0.15 + 0.0
Weighted ERM
0.8052 + 0.0073
AP
0.028 0.006
0.028 + 0.006
-0.0281 + 0.006
Constrained
0.799 + 0.009
AP
0.009 + 0.006
0.009 + 0.006
0.009 + 0.006
Reduction
0.723 = 0.0341
AP
0.0367 = 0.0172
0.0368 + 0.0172
-0.0367 = 0.0172
FairGrad
0.8042 = 0.0046
AP
0.0048 + 0.0033
0.004S 0.0033
-0.0048 = 0.0032
Unconstrained
0.8071 + 0.0072
Eodds
0.0212 + 0.0018
0.0322 + 0.009
-0.0256 + 0.0052
Constant
0.522 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.8074 = 0.0074
Eodds
0.0213 + 0.002
0.032 + 0.0086
-0.0254 + 0.0051
Constrained
0.79 1: 0.005
Eodds
0.005 0.003
0.009 1 0.005
0.002 + 0.002
BiFair
0.804 0.008
Eodds
0.021 0.003
0.025 1 0.004
-0.033 +: 0.01
FairBatch
0.809 + 0.0096
Eodds
0.018 + 0.0016
0.0262 + 0.0039
-0.0211 + 0.004
Reduction
0.6716 + 0.0251
Eodds
0.0226 + 0.006
0.0333 0.0107
-0.0404 0.0213
FairGrad
0.7978 = 0.0064
Eodds
0.0053 + 0.0019
0.007 + 0.0019
-0.009 + 0.0049
Unconstrained
0.8129 + 0.0021
Eopp
0.0075 + 0.0034
0.0107 + 0.0049
-0.0193 + 0.0086
Constant
0.524 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 0.0
Weighted ERM
0.8077 +:2 0.0078
Eopp
0.0076 1: 0.0034
0.011 +: 0.0049
-0.0196 + 0.0087
Constrained
0.814 0.003
Eopp
0.003 0.002
0.007 + 0.006
# 0.0
BiFair
0.808 + 0.01
Eopp
0.005 + 0.005
0.008 + 0.007
-0.012 + 0.012
FairBatch
0.81-19 + 0.0117
Eopp
0.0031 + 0.0014
0.0044 0.002
-0.0079 + 0.0036
Reduction
0.7397 + 0.0176
Eopp
0.026 + 0.0058
0.0669 + 0.0149
-0.0372 + 0.0083
FairGrad
0.8144 = 0.0021
Eopp
0.004 0.0037
0.006 + 0.0052
-0.0099 = 0.0097
34

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Published in Transactions on Machine Learning Research (08/2023)
Table 15: Results for the Dutch dataset with Non Linear Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.7937 0.0052
AP
0.0252 + 0.0091
0.0252 + 0.009
-0.0252 + 0.0091
Constant
0.524 + 0.0
AP
0.151 = 0.0
0.152 + 0.0
0.15 = 0.0
Weighted ERM
0.7954 + 0.0023
AP
0.0257 + 0.0089
0.0257 + 0.0089
-0.0257 = 0.0089
Adversarial
0.7939 + 0.0043
AP
0.0232 + 0.0071
0.0232 + 0.0071
-0.0232 + 0.007
Reduction
0.7421 + 0.0168
AP
0.0227 + 0.0141
0.0227 + 0.0142 -0.0227 + 0.0141
FairGrad
0.8043 0.0071
AP
0.0052 0.0026
0.0052 + 0.0026 -0.0052 = 0.0026
Unconstrained
0.7914 + 0.006
Eodds
0.0162 + 0.0062
0.0193 + 0.0071 -0.0263 + 0.0142
Constant
0.522 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.7958 0.0027
Eodds
0.0168 0.0053
0.0202 + 0.0048
-0.0261 = 0.0131
Adversarial
0.7928 + 0.0077
Eodds
0.0148 = 0.0041
0.0202 + 0.0066
-0.0211 = 0.006
Bilair
0.819 1 0.003
Eodds
0.021 + 0.004
0.03 +:: 0.005
-0.028 0.007
FairBatch
0.8091 + 0.012
Eodds
0.018 + 0.0021
0.0254 + 0.0058
-0.0248 1: 0.0062
Reduction
0.7144 0.0176
Eodds
0.0253 + 0.0073
0.0347 + 0.0123
-0.0323 = 0.0064
FairGrad
0.8013 + 0.0073
Eodds
0.0069 + 0.0031
0.0099 + 0-0038 -0.0095 + 0.0068
Unconstrained
0.8149 +:2 0.0031
Eopp
0.0055 +: 0.0024
0.0079 +: 0.0035
-0.014 + 0.0061
Constant
0.524 + 0.0
Eopp
0.0 = 0.0
0.0 = 0.0
0.0 + 0.0
Weighted ERM
0.8179 + 0.0044
Eopp
0.0066 + 0.0026
0.0095 + 0.0037
-0.017 0.0065
Adversarial
0,8156 0.0038
Eopp
0.004 + 0.0039
0.0058 + 0.0057
-0.0102 + 0.01
BiFair
0.819 1 0.003
Eopp
0.009 + 0.002
0.012 + 0.003
-0.022 0.006
FairBatch
0.8174 + 0.0031
Eopp
0.002 + 0.0012
0.0029 + 0.0017 -0.0052 + 0.0031
Reduction
0.7571 + 0.0061
Eopp
0.0219 + 0.0021
0.0563 + 0.0054
-0.0313 + 0.0028
FairGrad
0.8158 + 0.0051
Eopp
0.0036 + 0.0031
0.0051 + 0.0045
-0.0092 + 0.0079
35

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Published in Transactions on Machine Learning Research (08/2023)
0.10
FairGrad
0.10
FairGrad
0.10
FairGrad
0.09
Unconstrained
FairBatch
Opportunity
0.00
FairBatch
Accuracy Parity
0.08
Weighted
Equal Odds
0.08
Unconstrained
0.05
Uncanstrained
0.07
Reduction
0.07
Weighted Erm
0.07
Weighted Emn
0,06
Constrainnt
0.06
Reduction
0.06
Reduction
01.15
0.05
Constraints
0.005
Constraints
BiFair
BiFair
0,03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.67 Accuracy
0.75
0.59
0.67
0.71
0.79
0.63
0.67
0.71
0.75
0.79
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOOp
0.10
FairGrad
0.10
FairGrad
0.10
FairGrad
Accuracy Parity
0.09
Uncomtrainned
Equal Opportunity
0.00
FairBatch
0.08
Adversarial
0.08
Unconstrained
0.08
Unconstralned
0.07
Weighted Erm
Equal Odds
0.07
Adversarial
0.07
Adversarial
0.00
Reduction
0.06
Weighted Erm
0.00
Weighted
0.05
0.05
Reduction
0.05
Reduction
0.00
BiFair
0.004
BäFair
0.03
0.03
0.00
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.67 Accuracy
0.59
0.67
0.71
0.63
0.71
Accuracy
Accuracy
(d) Non Linear AP
Non Linear EOdds
(f) Non Linear EOpp
Figure 12: Results for the German dataset with different. fairness measures.
Table 16: Results for the German dataset with Lincar Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1 MAXIMUM
MINIMUM
Unconstrained
0.692 0.0232
AP
0.0226 + 0.0181
0.0169 0.0111 -0.0284 + 0.0256
Constant
0.73 + 0.0
AP
0,05 + 0.0
0.069 + 0.0
0.031 = 0.0
Weighted ERM
0.707 0.0344
AP
0.0243 + 0.0191
0.0186 0.0113
-0.0299 + 0.027
Constrained
0.733 + 0.033
AP
0.024 + 0.025
0.032 + 0.033
0.015 + 0.017
Reduction
0.631 0.0396
AP
0.0323 + 0.0139
0.0286 + 0.0202
-0.036 + 0.0185
FairGrad
0.744 + 0.0357
AP
0.0274 + 0.0212
0.0215 0.0123
-0.0334 - 0.0306
Unconstrained
0.69 + 0.0266
Eodds
0.0316 + 0.0207
0.0499 + 0.0341
-0.0618 + 0.0471
Constant
0.7 = 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.709 + 0.0296
Eodds
0.0324 + 0.0338
0.0461 0.046
-0.055 + 0.0626
Constrained
0.739 0.027
Eodds
0.037 0.012
0.072 1: 0.025
0.01 + 0.004
BiFair
0.698 0.039
Eodds
0.033 1 0.01
0.052 1. 0.023
-0.059 +: 0.029
FairBatch
0.7 + 0.0247
Eodds
0.0706 + 0.0184
0.1102 + 0.0489
-0.1134 + 0.0518
Reduction
0.707 0.0335
Eodds
0.0361 + 0.0175
0.0716 0.056
-0.0576
0.0266
FairGrad
0.734 + 0.0358
Eodds
0.0464 + 0.0201
0.0784 + 0.0232
-0.0721 + 0.0496
Unconstrained
0.704 + 0.0193
Eopp
0.0053 + 0.0035
0.0096 + 0.004
-0.0116 + 0.0117
Constant
0.7 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.706 + 0.0328
Eopp
0.0048 1: 0.0039
0.0097 + 0.0091
-0.0096 + 0.0092
Constrained
0.741 0.019
Eopp
0.005 0.002
0.015 1 0.006
0.0
BiFair
0.703 + 0.037
Eopp
0.007 + 0.006
0.014 + 0.015
-0.013 + 0.015
FairBatch
0.718 0.0229
Eopp
0.0172 + 0.0124
0.0272 + 0.0187
-0.0416 + 0.0396
Reduction
0.717 + 0.0441
Eopp
0.0183 + 0.014
0.036 + 0.0254
-0.0372 + 0.0407
FairGrad
0.723 + 0.0425
Eopp
0.0125 + 0.0043
0.0212 + 0.0087
-0.0288 + 0.0162
36

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Published in Transactions on Machine Learning Research (08/2023)
Table 17: Results for the German dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY; 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.695 + 0.0122
AP
0.0426 + 0.0241
0.0314 + 0.0144
-0.0537 + 0.0345
Constant
0.73 + 0.0
AP
0.05 = 0.0
0.069 + 0.0
0.031 0.0
Weighted ERM
0.703 + 0.0183
AP
0.035 = 0.0237
0.0265 + 0.0138
-0.0436 = 0.0338
Adversarial
0.681 + 0.0156
AP
0.041 + 0.0254
0.0327 + 0.0165
-0.0492 + 0.0368
Reduction
0.666 + 0.0198
AP
0.0173 + 0.0171
0.0131 + 0.0115 -0.0215 + 0.0231
FairGrad
0.714 + 0.026
AP
0.037 0.0222
0.0291 + 0.0119 -0.0448 = 0.0331
Unconstrained
0.689 + 0.0213
Eodds
0.0089 + 0.0052
0.0117 + 0.0045 -0.0144 + 0.0116
Constant
0.7 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.703 + 0.034
Eodds
0.0211 0.0106
0.0305 + 0.0186
-0.0372 = 0.0158
Adversarial
0.684 + 0.0097
Eodds
0.0184 + 0.0122
0.0263 + 0.0201
-0.0339 = 0.0237
Bilair
0.725 1: 0.031
Eodds
0.016 1 0.015
0.021 0.018 -0.027 0.018
FairBatch
0.692 1 0.026
Eodds
0.0489 1 0.0382
0.0607 + 0.0446
-0,0882 1 0.0983
Reduction
0.706 + 0.0272
Eodds
0.0489 + 0.0217
0.0742 + 0.0266
-0.0717 = 0.051
FairGrad
0.695 + 0.0237
Eodds
0.0095 + 0.004
0.0121 + 0.0046 -0.0175 + 0.0076
Unconstrained
0.686 +: 0.0215
Eopp
0.0124 +: 0.0075
0.0227 +: 0.0128 -0.0269 +: 0.0227
Constant
0.7 # 0.0
Eopp
0.0 =00=:0 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.7 + 0.0261
Eopp
0.0066 + 0.0057
0.0131 + 0.0071
-0.0133 = 0.0173
Adversarial
0.687 + 0.0129
Eopp
0.0085 + 0.0051
0.0203 + 0.0147
-0.0137 + 0.0099
BiFair
0.727 + 0.023
Eopp
0.015 + 0.013
0.023 0.019
-0.036 0.038
FairBatch
0.697 + 0.025
Eopp
0.0084 + 0.0079
0.0235 + 0.0226
-0.0102 + 0.0094
Reduction
0.701 + 0.0397
Eopp
0.0102 + 0.008
0.0242 + 0.024
-0.0167 + 0.0134
FairGrad
0.696 + 0.0166
Eopp
0.0052 + 0.0038
0.0093 + 0.0064
-0.0115 + 0.0108
37

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Published in Transactions on Machine Learning Research (08/2023)
0.12
0.13
Fair Grad
0.12
0.13
FaiGGrad
0.12
0.13
FairGrad
0.11
Unconstrained
0.11
FairBatch
Opportunis
0.11
FairBatch
0.10
Weighted
0.10
Unconstrained
0.10
Uncanstrained
0,00
Reduction
Gual O4d
Weighted Erm
0.09
Weighted Emn
0.08
Constraints
Reduction
0.00
Reductioon
Constraints
0.06
Constraints
0.05
BiFair
0.05
BiFair
0.03
0.04
0.033
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.64 Accuracy
0.84
0.44
0 64
0.74
0.94
0.54
0.64
0.74
0.84
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.12
0.13
FairGrad
0.13
0.12
FairGrad
0.12
0.13
FairGrad
Accuracy Parity
0.11
Uncomttriined
0.11
Equal Opportunis
P0.11
FairBatch
0.10
Adversarial
0.10
Unconstrained
0.10
Unconstrained
0100
Weighted Erm
qual Odd
0.00
Adversarial
02.09
Advirsarial
0.08
0.08
0.08
3,00
Reduction
Weighted
0.07
Weighted
0.06
Reduction
0.06
Reduction
0.05
BiFair
0.05
BäFair
0.04
0.03
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.00
0.64
0.84
0.44
0.64
0.74
0.54
0.74
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
Non Linear EOdds
(f) Non Linear EOpp
Figure 13: Results for the Gaussian dataset with different. fairness measures.
Table 18: Results for the Gaussian datasct with Lincar Models. All the results are averaged over 5 runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.8689 + 0.0037
AP
0.0966 + 0.0029
0.0957 0.0028
-0.0974 + 0.0036
Constant
0.497 + 0.0
AP
0,001 + 0.0
0.001 + 0.0
0.001 = 0.0
Weighted ERM
0.869 0.0039
AP
0.0966 + 0.0026
0.0957 0.0023
-0.0974 + 0.0034
Constrained
0.799 + 0.004
AP
0.003 + 0.002
0.003 +: 0.002
0.003 + 0.002
Reduction
0.7891 = 0.0266
AP
0.0575 + 0.0114
0.057 = 0.0118
-0.0579 = 0.0111
FairGrad
0.8516 = 0.0064
AP
0.0558 + 0.0094
0.0553 0.0093
-0.0562 = 0.0096
Unconstrained
0.869 + 0.0037
Eodds
0.0971 + 0.0026
0.1872 + 0.0067
-0.1896 + 0.0056
Constant
0.499 = 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.869 0.0039
Eodds
0.0971 + 0.0023
0.1869 0.0063
-0.1894 = 0.0051
Constrained
0.497 + 0.003
Eodds
0.0 + 0.0
0.0 1 0.0
0.0 1 0.0
BiFair
0.873 0.004
Eodds
0.113 1 0.004
0.21 1 0.007
-0.213 1. 0.004
FairBatch
0.8649 + 0.0025
Eodds
0.0902 + 0.0035
-0.1719 + 0.0079 0.0732 0.0198
0.1717 + 0.0046
Reduction
0.6241 + 0.054
Eodds
0.0632 + 0.0164
-0.074 + 0.0226
FairGrad
0.8459 = 0.01
Eodds
0.0786 + 0.0051
0.1504 + 0.0102
-0.1527 + 0.0142
Unconstrained
0.8598 + 0.0121
Eopp
0.0928 + 0.0012
0.1845 + 0.0041
-0.1869 + 0.0041
Constant
0.498 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 0.0
Weighted ERM
0.8599 +: 0.0121
Eopp
0.0931 +: 0.0011
0.1849 + 0.004
-0.1874 + 0.004
Constrained
0.698 0.005
Eopp
0.004 0.002
0.008 0.005
0.0 0.0
BiFair
0.863 + 0.009
Eopp
0.1 + 0.003
0.2 + 0.007
-0.202 + 0.006
FairBatch
0.8635 + 0.0024
Eopp
0.085 0.0023
0.17 0.0032
-0.1702 + 0.0065
Reduction
0.6251 + 0.0355
Eopp
0.0189 + 0.0138
0.0379 + 0.0271
-0.0378 + 0.0282
FairGrad
0.8431 + 0.0065
Eopp
0.0752 + 0.0043
0.1494 + 0.0087
-0.1514 + 0.0094
38

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Published in Transactions on Machine Learning Research (08/2023)
Table 19: Results for the Gaussian dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY; 1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.88 0.0038
AP
0.0897 + 0.0045
0.0888 + 0.0035 -0.0905 + 0.0055
Constant
0.497 + 0.0
AP
0.001 + 0.0
0.001 + 0.0
0.001 0.0
Weighted ERM
0.8809 + 0.0048
AP
0.0903 + 0.0045
0.0894 + 0.0033
-0.0911 = 0.0057
Adversarial
0.8725 + 0.0115
AP
0.0858 + 0.0077
0.0851 + 0.0076
-0.0866 + 0.0081
Reduction
0.718 + 0.0251
AP
0.0694 + 0.0237
0.0699 + 0.0236 -0.0689 + 0.0239
FairGrad
0.8542 0.0047
AP
0.0352 0.0047
0.0349 + 0.0048 -0.0355 = 0.0046
Unconstrained
0.8814 0.0024
Eodds
0.093 + 0.0032
0.1807 + 0.0066 -0.183 0.005
Constant
0.499 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.8821 0.0031
Eodds
0.0939 0.0013
0.1826 + 0.0042
-0.185 0.0033
Adversarial
0.8775 + 0.0091
Eodds
0.0852 + 0.007
0.1643 = 0.0125
-0.1666 = 0.0146
Bilair
0.868 1 0.013
Eodds
0.092 + 0.011
0.167 T 0.035
-0.168 0.031
FairBatch
0.8735 1 0.0032
Eodds
0.0749 1 0.0041
0.1455 1 0.0059
-0.1456 1 0.0056
Reduction
0.7309 0.0189
Eodds
0.0262 + 0.0141
0.0438 + 0.0257
-0.0435 = 0.0265
FairGrad
0.8539 + 0.0056
Eodds
0.0596 + 0.0068
0.1013 + 0.0147 -0.1025 + 0.0144
Unconstrained
0.8801 +:2 0.004
Eopp
0.0902 +: 0.0017
0.1792 + 0.0011 -0.1816 +: 0.0053
Constant
0.498 + 0.0
Eopp
0.0 = 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.8805 + 0.0046
Eopp
0.0912 + 0.0008
0.1812 + 0.0024
-0.1837 = 0.0045
Adversarial
0.8754 0.0086
Eopp
0.0806 +: 0.0066
0.1605 1 0.0128
-0.1628 1 0.0143
BiFair
0.88 + 0.003
Eopp
0.086 + 0.005
0.17 + 0.013
-0.172 0.009
FairBatch
0.874 + 0.0035
Eopp
0.0733 + 0.0029
0.1465 + 0.0054
-0.1467 + 0.0066
Reduction
0.6868 + 0.0234
Eopp
0.0505 + 0.0179
0.1015 + 0.0359
-0.1005 + 0.036
FairGrad
0.8543 + 0.0082
Eopp
0.0517 + 0.0095
0.1028 + 0.0191
-0.1011 + 0.0192
39

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Published in Transactions on Machine Learning Research (08/2023)
FairGrad
FairGrad
FairGrad FairBatch
Unconstrained
FairBatch
Equal Opportunity
Accuracy Parity
0.15
Weighted
Equal Odds
0.15
Unconstrained
0.15
Reduction
Weighted Erm
Weighted Emn
0..1 0.10
Constraints
0.10
Reduction
0.10
Reduction
BiFair
BiFair
0.05
0.05
0.05
0.00
0.00
0.00
0.57
0.61
0.65
0.69
0.73
0.77
0.57
0.61
0.65
0.59
0.73
0.77
0.57
0.61
0.65
0.69
0.73
0.77
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.20
0.20
0.20
FairGrad
FairGrad
FairGrad
Unconstrained
FairBatch
Opportunity
FairBatch
Accuracy Parity
0.15
Equal Odds
Unconstrained
Unconstrained
Weighted
Adversarial
Adversarial
Reduction
Weighted Erm
Weighted Enn
0.10
0.10
Reduction BiFair
Reduction BiFair
Equal
0.05
0.05
0.05
0.00
0.00
0.00
0.57
0.61
0.65
0.69
0.73
0.77
0.57
0.61
0.65
0.69
0.73
0.77
0.57
0.61
0.65
0.69
0.73
0.77
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear F.Odss
(f) Non Linear E.ppp
Figure 14: Results for the Twitter Sentiment dataset with different fairness mcasures.
Table 20: Results for the Twitter Sentiment dataset with Lincar Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY
MEASURE MEAN ABS. 1 MAXIMUM
MINIMUM
Unconstrained
0.7211 + 0.004
AP
0.0426 + 0.0011
0.0426 0.0011
-0.0426 0.0011
Constant
0.5 + 0.0
AP
0,0 + 0.0
0.0 # 0.0
0.0 = 0.0
Weighted ERM
0.7212 + 0.0044
AP
0.0426 + 0.0011
0.0126
0.0011
-0.0426 + 0.0011
Constrained
0.72 + 0.002
AP
0.04 + 0.003
0.04 + 0.003
0.01 + 0.003
Reduction
0.6008 = 0.022
AP
0.0159 + 0.0092
0.0159
0.0092
-0.0159 = 0.0092
FairGrad
0.7219 = 0.0027
AP
0.0462 + 0.0021
0.0462 0.0021
-0.0462 = 0.0021
Unconstrained
0.7237 + 0.0054
Eodds
0.1867 + 0.0052
0.2287 + 0.0078
-0.2288 + 0.0078
Constant
0.5 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.7234 = 0.0054
Eodds
0.188 + 0.0033
0.2314 0.0056
-0.2315 + 0.0056
Constrained
0.72 1: 0.004
Eodds
0.012 1 0.002
0.019 1 0.005
0.006 + 0.005
BiFair
0.736 0.009
Eodds
0.041 0.012
0.056 1 0.022
-0.556 +: 0.022
FairBatch
0.7413 + 0.0014
Eodds
0.1391 + 0.0043
0.1755 + 0.0084
-0.1756 + 0.0084
Reduction
0.5962 + 0.0113
Eodds
0.0213 + 0.0108
0.0314 0.0211
-0.0314
0.021
FairGrad
0.7193 = 0.0062
Eodds
0.0154 + 0.0051
0.0204 + 0.0098
-0.0204 + 0.0098
Unconstrained
0.7244 + 0.0051
Eopp
0.0719 + 0.0012
0.1439 + 0.0023
-0.1438 + 0.0023
Constant
0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.72 + 0.0054
Eopp
0,0718 1: 0.0013
0.1437 + 0.0026
0.0 # -0.1436 + 0.0026
Constrained
0.752 0.004
Eopp
0.002 0.001
0.005 1
BiFair
0.746 + 0.009
Eopp
0.009 + 0.004
0.017 + 0.009
-0.017 + 0.009
FairBatch
0.7426 + 0.001
Eopp
0.0429 + 0.0005
0.0858 0.0011
-0.0858 + 0.001
Reduction
0.6381 + 0.0039
Eopp
0.0712 + 0.0117
0.1421 + 0.0231
-0.1225 + 0.0231
FairGrad
0.7518 = 0.0069
Eopp
0.0024 + 0.002
0.0049 + 0.004
-0.0049 + 0.004
40

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Published in Transactions on Machine Learning Research (08/2023)
Table 21: Results for the Twitter Sentiment dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY;
1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.715 + 0.0043
AP
0.0392 + 0.0055
0.0392 + 0.0055 -0.0392 + 0.0055
Constant
0.5 + 0.0
AP
0.0 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.7183 + 0.0042
AP
0.0427 + 0.0019
0.0427 + 0.0019 -0.0427 = 0,0019
Adversarial
0.7385 + 0.0075
AP
0.0367 + 0.0027
0.0367 + 0.0027 -0.0368 + 0.0027
Reduction
0.6555 0.0162
AP
0.0101 + 0.0038
0.0101 + 0.0038 -0.0101 + 0.0038
FairGrad
0.7154 0.0047
AP
0.0368 0.0079
0.0367 + 0.0078 -0.0368 = 0.0079
Unconstrained
0.7167 + 0.0126
Eodds
0.1854 + 0.0061
0.2349 + 0.0091 -0.235 0.0091
Constant
0.5 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.718 + 0.0137
Eodds
0.1882 0.0062
0.2379 + 0.0073
-0.2381 = 0.0073
Adversarial
0.7393 + 0.0024
Eodds
0.0382 = 0.0056
0.06 = 0.0151
-0.06 = 0.0151
Bilair
0.01
Eodds
0.039 1 0.016
0.058 0.017
-0.058 0.017
FairBatch
0.7318 1 0.004
Eodds
0.1313 1: 0.0057
0.1724 1 0.0055
-0.1725 1 0.0055
Reduction
0.6653 0.0134
Eodds
0.0133 + 0.0097
0.0199 + 0.0172
-0.0199 = 0.0173
FairGrad
0.717 + 0.0082
Eodds
0.0109 + 0.0027
0.0165 + 0.0053
-0.0165 + 0.0053
Unconstrained
0.7147 + 0.01188
Eopp
0.0653 +: 0.0062
0.1306 + 0.0124 -0.1306 +: 0.0124
Constant
0.5 + 0.0
Eopp
0.0 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.7074 + 0.0158
Eopp
0.0672 + 0.0062
0.1346 + 0.0125
-0.1345 = 0.0125
Adversarial
0.7471 + 0.0042
Eopp
0.005 + 0.0035
0.0099 1 0.007
-0.0099 1 0.007
BiFair
0.747 + 0.009
Eopp
0.007 + 0.005
0.013 0.01
-0.013 0.01
FairBatch
0.7359 + 0.0011
Eopp
0.0368 + 0.0012
0.0736 + 0.0025
-0.0736 + 0.0025
Reduction
0.681 + 0.0078
Eopp
0.0436 + 0.0071
0.0871 + 0.0143
-0.0871
+ 0.0143
FairGrad
0.7401 + 0.0059
Eopp
0.0049 + 0.0041
0.0099 + 0.0083 -0.0099 + 0.0083
41

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Published in Transactions on Machine Learning Research (08/2023)
Accuracy Parity
0.10
0.11
0.12
Weighted Em
Unconstrained
0..0
0.1]
0.12
Unconstrained
FairBatch
2... 0.11
Opportunity
0.10
0.12
FairBatch
Unconstrained
0.00
[qual Odds
0.00
0.09
0108
Reduction
0.08
Weighted Erm
0.08
Weighted
Constraints
Reduction
Reduction
0.00
Constraints
0.06
Constraints
0.05
0.05
0.05
0.04
0.04
0.04
01033
0.03
0.003
0.02
0.02
0.02
0.01
0.01
0.01
0.00
0.00
0.000
0.35
0.47
0.58
0.70
0.81
0.93
0.35
0.47
0.58
0.70
0.81
0.93
0.35
0.47
0.58
0.70
0.81
0.93
Accuracy
Accuracy
Accuracy
(a) Linear AP
(b) Linear EOdds
(c) Linear EOpp
0.13
0.13
0.13
0.12
FairGrad
0.12
FairGrad
0.12
FairGrad
Accuracy Parity
0.10
0.11
Adversarial
Uncomstrained
0.11
FairBatch
Unconstrained
Equal Opportunit
20.1
0.10
FairBatch
Uncanstrained
0.08
Weighted Er
[qual Odds
Adversaria
0.08
Adversarial
0.07
0.07
Weighted
0.07
Weighted
0108
0.06
Reduction
0.06
Reduction
0.05
0.05
BiFair
0.05
BiFair
0.00
0.04
0..04
0.00
0.03
0.03
0.01
0.01
0.00
0.00
0.00
0.35
0.58
0.35
0.70
0.93
0.70
Accuracy
Accuracy
Accuracy
(d) Non Linear AP
(e) Non Linear EOdds
(f) Non Linear EOpp
Figure 15: Results for the Polktables Adult dataset with different fairness measures.
Table 22: Results for the Folktables Adult datasct with Lincar Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (L) ACCURACY 1
MEASURE MEAN ABS. 1
MAXIMUM
MINIMUM
Unconstrained
0.7905 + 0.0033
AP
0.0131 + 0.0021
0.0123 0.0021
-0.0138 + 0.0022
Constant
0.666 + 0.0
AP
0,053 + 0.0
0.056 + 0.0
0.051 = 0.0
Weighted ERM
0.7906 + 0.0032
AP
0.0127 + 0.0023
0.0119
0.0022
-0.0134 + 0.0024
Constrained
0.467 + 0.115
AP
0.036 + 0.003
0.039 + 0.003
0.031 + 0.003
Reduction
0.733 = 0.0106
AP
0.0653 + 0.0114
0.0614 0.011
-0.0692 = 0.0118
FairGrad
0.7837 = 0.0049
AP
0.0023 = 0.0009
0.0023 + 0.001
-0.0022 = 0.0008
Unconstrained
0.789 + 0.0026
Eodds
0.0301 + 0.011
0.0377 + 0.0153 -0.0458 + 0.0184
Constant
0.667 = 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 = 0.0
Weighted ERM
0.7886 = 0.0032
Eodds
0.0294 + 0.012
0.0364 0.0169
-0.0443 = 0.0206
Constrained
0.663 + 0.032
Eodds
0.008 1 0.003
0.013 1: 0.004
0.004 + 0.002
BiFair
0.768 0.007
Eodds
0.008 0.005
0.011 1 0.006
40.11 +: 0.008
FairBatch
0.788 + 0.0027
Eodds
0.0045 + 0.0033
0.0069 + 0.0065
-0.0063 + 0.0049
Reduction
0.6922 + 0.0346
Eodds
0.077 0.0322
0.0761 0.0257
-0.0903
0.0378
FairGrad
0.7885 = 0.0027
Eodds
0.0043 + 0.0019
0.0073 + 0.0037
-0.0068 + 0.0045
Unconstrained
0.7902 + 0.0038
Eopp
0.0094 + 0.0031
0.0162 + 0.0053
-0.0215 + 0.0071
Constant
0.667 0.0
Eopp
0.0 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.7893 +:2 0.0031
Eopp
0.009 1: 0.003
0,0155 + 0.0051
-0.0206 + 0.0069
Constrained
0.706 0.002
Eopp
0.004
0.01 0.001
0.0 # 0.0
BiFair
0.77 + 0.002
Eopp
0.019 + 0.01
0.033 + 0.017
-0.044 + 0.023
FairBatch
0.79 + 0.0031
Eopp
0.0012 + 0.0015
0.0022 0.0026
-0.0026 + 0.0034
Reduction
0.7388 + 0.0144
Eopp
0.0409 + 0.0111
0.0932 + 0.025
-0.7701 + 0.0194
FairGrad
0.7893 = 0.0026
Eopp
0.0011 + 0.0009
0.0024 + 0.002
-0.0021 = 0.0016
42

---

Published in Transactions on Machine Learning Research (08/2023)
Table 23: Results for the Folktables Adult dataset with Non Linear Models. All the results are averaged over runs. Here MEAN ABS., MAXIMUM, and MINIMUM represent the mean absolute fairness value, the fairness level of the most well-off group, and the fairness level of the worst-off group, respectively.
FAIRNESS
METHOD (NL) ACCURACY;
1
MEASURE MEAN ABS.
MAXIMUM
MINIMUM
Unconstrained
0.8037 + 0.0037
AP
0.0131 + 0.0017
0.0123 + 0.0016 -0.0139 + 0.0017
Constant
0.666 + 0.0
AP
0.053 + 0.0
0.056 + 0.0
0.051 = 0.0
Weighted ERM
0.8046 0.0049
AP
0.0131 + 0.0014
0.0123 + 0.0014
-0.0138 = 0,0015
Adversarial
0.8016 0.0053
AP
0.0122 + 0.0016
0.0115 + 0.0015
-0.0129 + 0.0016
Reduction
0.7293 + 0.0133
AP
0.0991 + 0.0149
0.0932 + 0.0139 -0.1051 + 0.016
FairGrad
0.7917 0.0025
AP
0.0016 0.0011
0.0016 + 0.0011 -0.0016 = 0.001
Unconstrained
0.7947 0.0078
Eodds
0.0314 + 0.0059
0.0373 + 0.0058 -0.0454 + 0.0066
Constant
0.667 + 0.0
Eodds
0.0 = 0.0
0.0 + 0.0
0.0 + 0.0
Weighted ERM
0.7902 0.0049
Eodds
0.0327 0.0061
0.04 + 0.0067
-0.0488 0.0077
Adversarial
0.806 + 0.0047
Eodds
0.0035 + 0.0018
0.0051 = 0.0021
-0.0053 = 0.0028
Bilair
0.793 1 0.006
Eodds
0.006 + 0.003
0.007 0.003
-0.007
0.004
FairBatch
0.8061 1: 0.0044
Eodds
0.0051 +: 0.0015
0.0087 + 0.0048
-0.0084 1 0.0029
Reduction
0.7416 0.01
Eodds
0.0933 + 0.022
0.1517 + 0.0311
-0.1244 = 0.026
FairGrad
0.7997 + 0.0087
Eodds
0.0045 + 0.0029
0.0067 + 0.0045
-0.0071 + 0.0058
Unconstrained
0.7902 +: 0.0044
Eopp
0.0097 +: 0.0026
0.0168 +:: 0.0045 -0.0222 +: 0.006
Constant
0.667 + 0.0
Eopp
0.0 = 0.0
0.0 = 0.0
0.0 0.0
Weighted ERM
0.7947 + 0.0022
Eopp
0.0105 + 0.0027
0.0181 + 0.0047
-0.024 0.0062
Adversarial
0.8108 0.0161
Eopp
0.0034 + 0.0057
0.0041 + 0.0057
-0.0095 + 0.017
BiFair
0.793 1 0.008
Eopp
0.028 + 0.017
0.048 0.029
-0.064 0.039
FairBatch
0.8038 + 0.0063
Eopp
0.0008 + 0.0005
0.0014 + 0.0009
-0.0018 + 0.0012
Reduction
0.7334 0.0155
Eopp
0.0573 + 0.0116
0.1307 + 0.0263
-0.0986 + 0.0199
FairGrad
0.8058 + 0.0035
Eopp
0.0014 +: 0.0014
0.003 + 0.0031
-0.0026 + 0.0024
43


Reviews:
[{"review_id": "XrQ69h6Rpm", "reviewer": "Reviewer_C7Px", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "S1) The empirical results are thorough and show extensive results with four datasets, (up to) five baselines, and four fairness metrics. The empirical results on batch size and epsilon values is helpful for better understanding the effects of these parameters.\n\nW1) My biggest question of the paper is that it's is unclear where to place FairGrad within the pantheon of existing fairness algorithms. As explained in Section 4.4, there is no fairness method that is consistently better than the others over all of different datasets. Why should we consider FairGrad compared to the others? \n\nThe introduction claims that the \"complexity of the existing models\" is the main obstacle for existing methods. Ease of implementation is a key selling point for FairGrad, but it appears that the other models are merely missing an easily accessilbe code package? I had originally thought that FairGrad's strength would be that it could be used for off-the-shelf algorithms as opposed to having to know the loss function, but it seems that other methods have that as well.\n\nI had thought that FairGrad was more computationally efficient than other algorithms, hence one of its strengths. Despite Table 3 going in depth about computational speed, the paper doesn't mention FairGrad as being superior in runtime than other methods.\n\nSome clarify around this would be helpful.\n\nSmall comment that Algorithm 1 line 5 is italicized and it's unclear why.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents FairGrad, a method for inducing fairness in gradient-based algorithms. Fairness here is defined as loss functions compared across groups, and the FairGrad method reweights gradients based on the group membership of the associated individual. The paper presents empirical results on four datasets and show that the FairGrad method has comparable and at times better results than other baselines.\n", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "More guidance about when to use FairGrad compared to alternative fairness methods would strengthen the paper. As it stands, there are a variety of fairness methods who have similar accuracy-fairness tradeoffs. Highlighting what FairGrad contributes to the landscape will help the community leverage the method.\n", "broader_impact_concerns": "Existing Limitations and Social Impact section is sufficient.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "kGqdXga0Uf", "reviewer": "Reviewer_EMqe", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n(1) FairGrad is a simple gradient descent algorithm for solving a wide class of fairness constrained optimization problems. The experimental results suggest that it works well in a variety of settings and is computationally more efficient than competing methods.\n\nWeakness:\n\n(1) Throughout the paper, the authors assume \"we will assume that what was measured on our finite sample is sufficiently close to what\nwould be obtained if one had access to the overall distribution\". In other words, their analysis of FairGrad provides no guarantees about its generalization to the true underlying distribution F(). This stands in contrast to the literature on fairness-constrained optimization problems, where, for example, papers like Agarwal et al. (2018) and Agarwal et al. (2019) provide generalization bounds on their returned solutions (i.e., they are close to the true population optimum with high probability). \n\n(2) In light of (1), the paper can be thought of as just providing a particular optimization routine for the sampled data more in the spirit of the analysis in Cotter et al. (2019). However, here too the authors analysis is lacking -- in particular, Cotter et al. (2019) are able to provide some optimization guarantees about when their reductions based approach will in fact converge (i.e., how must the hyper-parameters be set, how many iterations must be run etc). ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a modification to gradient descent algorithms to incorporate/enforce fairness constraints. The procedure follows closely in spirit with some modifications to the reductions-based algorithms for solving fairness-constrained optimization problems in Agarwal et al. (2018) and Cotter et al. (2019).\n\nFor a particular class of fairness definitions, the authors consider solving a standard fairness constrained optimization problem that they re-interpret as an unconstrained problem with some lagrange multipliers. As in Agarwal et al. (2018) and Cotter et al. (2019), the authors propose to solve this unconstrained problem by searching for saddle points via an iterative search -- in each step, first the $\\lambda$-player moves to select $\\lambda$ to drive up the objective function, and then the model player updates the prediction function's weights to lower loss. The author's main insight is that for the class of fairness definitions considered in the paper, the min-player's objective function is just a particular weighted average of group-specific losses. As a result, we can also apply take gradient descent steps to optimize the min-player's objective function. Altogether, the author's main contribution effectively is to p ropose that we can solve fairness constrained optimization problems by applying a gradient descent ascent algorithm with an objective function that updates at each step. Such algorithms have been extensively analyzed in a literature on adversarial learning such as Madry et al. (2019).\n\nReferences: \nMadry et al. (2019): Towards Deep Learning Models Resistant to Adversarial Attacks", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(1) I don't think it's fair to write ``They are also limited in the range of problems to which they can be applied. For example, the work\nof Agarwal et al. (2018) can only be applied in a binary classification setting...'' in the introduction. This a correct statement about the specific paper (Agarwal et al. 2018) but the broader in-processing literature certainly provides extensions of these algorithms to richer settings. See, for example, Agarwal et al. (2019) that extends these procedures to cases with a continuous outcome. By contrast, the main text of this paper focuses on a case with discrete outcomes.\n\n(2) In the main text, you should clarify that the group definitions Tk can also depend on the true label y. This is important with how you can nest definitions like equalized odds into the fairness definition.\n\n(3) Per the weaknesses above, I think the authors need to incorporate a more up-front discussion about how their analysis is limited relative to existing work on fairness-constrained optimization. \n\n(4) The authors should discuss how their proposed algorithm is closely connected to procedures that are popular in adversarial learning (e.g., see Madry et al. (2019) and related citations). I view the author's main contributions here as showing that we can apply the gradient descent-ascent optimization procedures that are popular in adversarial learning to do fairness constrained optimization over the particular combination of loss function and fairness definitions considered. \n\nReferences:\nAgarwal et al. (2019): Fair Regression: Quantitative Definitions and Reduction-based Algorithms", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "3cjM1xn4KB", "reviewer": "Reviewer_Akhq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main strengths of the paper are as follows: \n\n--The proposed approach is very simple.\n\n--The proposed approach can deal with multi-class classification, more than two subgroups, and can handle several notions of fairness. \n\n\nHowever, the paper suffers from some shortcomings:\n\n--Given the plethora of papers dealing with proposing new in-processing methods to provide group fairness, it is unclear how much the new approach can add to the existing approaches. In particular, the approach does not meaningfully outperform previous in-processing methods as shown in the experiments. \n\n-- Given that the major contribution of the paper is empirical, there are a few major issues with the experiments. In particular, some baselines are not included, and also some experimental details and rationale behind the choices are not clear in the paper (see the Requested Changes for more details).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a new in-processing approach, called FairGrad, to enforce statistical group fairness notions such as error parity, equality of opportunity, and equalized odds.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would appreciate it if the paper addresses the following issues:\n\n-- Are there any situations in which FairGrad outperforms the existing in-processing approach to achieve a better fairness/accuracy tradeoff? \n\n-- A major omitted baseline is the reduction approach of Agarwal et al., 2018 which is perhaps the most commonly used in-processing baseline in the literature. \n\n-- The paper misses some crucial discussion on in-processing approaches for an arbitrary number of groups e.g., https://proceedings.mlr.press/v80/hebert-johnson18a.html and https://proceedings.mlr.press/v80/kearns18a.html. \n\n-- Many of the experimental details are unclear. For example, what is the non-linear model used in the experiments? How does the performance change as a function of different model classes? What is the justification for using linear models? ", "broader_impact_concerns": "Broader impacts are addressed in a satisfactory manner. ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The three reviewers have positive views on this manuscript. The revised version has incorporated reviewers’ comments including adding experimental results related of Agarwal et al., 2018, and adding discussion on when the proposed method outperforms other existing in-processing approaches to achieve a better fairness/accuracy tradeoff.
Decision: Accept (poster)