# My Publication Title — Multiple Authors

Xuan Zhao1, Simone Fabbrizzi2, Paula Reyero Lobo3, Siamak Ghodsi4, Klaus
Broelemann1, Steffen Staab 5, Gjergji Kasneci 6

# Adversarial Reweighting Guided by Wasserstein Distance for Bias Mitigation

Xuan Zhao1, Simone Fabbrizzi2, Paula Reyero Lobo3, Siamak Ghodsi4, Klaus
Broelemann1, Steffen Staab 5, Gjergji Kasneci 6

###### Abstract

The unequal representation of different groups in a sample population can lead
to discrimination of minority groups when machine learning models make
automated decisions. To address these issues, fairness-aware machine learning
jointly optimizes two (or more) metrics aiming at predictive effectiveness and
low unfairness. However, the inherent under-representation of minorities in
the data makes the disparate treatment of subpopulations less noticeable and
difficult to deal with during learning. In this paper, we propose a novel
adversarial reweighting method to address such _representation bias_. To
balance the data distribution between the majority and the minority groups,
our approach deemphasizes samples from the majority group. To minimize
empirical risk, our method prefers samples from the majority group that are
close to the minority group as evaluated by the Wasserstein distance. Our
theoretical analysis shows the effectiveness of our adversarial reweighting
approach. Experiments demonstrate that our approach mitigates bias without
sacrificing classification accuracy, outperforming related state-of-the-art
methods on image and tabular benchmark datasets.

## Introduction

Machine learning models that are trained on personal data may discriminate
against groups with sensitive attributes. Broadly speaking, there are three
major paradigms to address this problem. The first paradigm assumes fairness
can be measured. Then, the minimization of unfairness metrics is integrated in
the empirical risk minimization as a multi-objective optimization problem
(Aghaei, Azizi, and Vayanos 2019; Berk et al. 2017). The second paradigm
assumes that discrimination arises from the use of protected (i.e., sensitive)
attributes and those correlated to them. Therefore, removing sensitive
information from the input data can support learning fair models (Creager et
al. 2019). The third paradigm builds on the assumption that discrimination
arises from biased labeling processes (e.g., through biased domain knowledge
or biased human feedback). Corresponding approaches aim at identifying and
correcting label bias (Jiang and Nachum 2019), such as the adaptive sensitive
reweighting of instances (Krasanakis et al. 2018).

These paradigms do not deal directly with the issue that, by definition,
minority groups are smaller than the majority. The effects of under-
represented data samples in the learning process are ‘overridden’ by the
prevalence of data samples from the majority group. The under-representation
negatively affects the sensitivity of the fairness metrics and can hide
undesirable correlations between attributes in the minority group. That is,
leading to a _representation bias_.

We propose a reweighting scheme to mitigate predictive quality issues arising
from the imbalance between sensitive groups. We do so by mapping the data into
a latent space where the data distribution becomes non-discriminatory with
respect to the sensitive attribute. Simultaneously, the empirical risk for the
classification task at hand is minimized. Our method addresses representation
bias by weighting the samples from the majority group. It aims to maintain the
class-wise discriminatory information of the data samples from the majority
group that are further away from the minority group, but downplay their
importance. Hence, the majority and minority groups become similar in
distribution and almost non-discriminatory in classification. We use the
critic of a Wasserstein Generative Adversarial Network (WGAN) with gradient
penalty (Gulrajani et al. 2017) to approximate distances between samples from
the minority and reweighted majority groups in the latent space.

The rationale for our method is that if subgroups are sufficiently represented
in a non-discriminatory way, bias in prediction would be substantially
reduced, if not eliminated (Chai and Wang 2022). Reweighting instances has
been adopted in methods for learning from imbalanced datasets (Bao et al.
2020; Zhang et al. 2019), which focus on optimizing the performance under a
class imbalance, without considering representation bias. Our method is
different from existing adversarial methods (Adel et al. 2019; Madras et al.
2018; Wadsworth, Vera, and Piech 2018) in exploiting the competition between
the reweighing component and the discriminator of the GAN framework, as an
additional discriminator has been generally used to decorrelate feature
embeddings from sensitive information.

We perform experiments on different datasets and compare with four state-of-
the-art fairness-aware methods. Our method outperforms its competitors in
mitigating bias while maintaining high prediction quality, as demonstrated by
the experimental evaluation of image and tabular benchmark datasets. Hence,
our method inherently addresses fairness as well as prediction quality issues
that might arise from learning on imbalanced datasets with respect to
sensitive groups.

We summarize our contribution as follows: (1) We formulate a novel data
transformation and sample-based reweighting method for mitigating
representation bias related to sensitive groups in classification tasks. (2)
We show theoretically that by closing the Wasserstein distance gap between
sensitive groups in the latent space during training, our reweighting approach
leads to predictions that adhere to demographic parity. (3) We provide a
thorough evaluation of the proposed technique on image and tabular benchmark
datasets and show the viability of our approach with respect to robustness to
fairness, accuracy and label noise. Code is available at
https://anonymous.4open.science/r/wasserstein˙reweight-46E6/.

## Related Work

#### Adversarial methods

Existing adversarial fairness methods (Adel et al. 2019; Madras et al. 2018;
Wadsworth, Vera, and Piech 2018) use in an in-processing fashion a
discriminator to decorrelate the embeddings and the sensitive attribute. The
authors of (Choi et al. 2019) and (Kim et al. 2019) propose to minimize mutual
information between the biased labels and the embedding through adversarial
training. These works disentangle the sensitive attribute in the latent space,
yet they do not consider the under-representation of sensitive groups. Our
work considers representation bias in the decorrelation process by reweighting
to align the distributions of the sensitive groups instead of only adjusting
the encoder.

#### Reweighting methods

Fairness with Adaptive Weights (Chai and Wang 2022) also constrains the sum of
weights among sensitive groups to be equal, assigning weights to a sample
based on its misclassification likelihood. Adaptive sensitive reweighting to
mitigate bias (Krasanakis et al. 2018) assigns weights to samples based on
their alignment with the unobserved true labeling. As highlighted example,
Adversarial reweighting for domain adaptation (Gu et al. 2021) aims to align
the distributions of the source and target domains, yet it deals with the
domain adaptation problem. In our work, we extend the concept of reweighting
based on the Wasserstein distance to the fairness domain.

#### Imbalanced classification

There are two main imbalanced classification methods: resampling and cost-
sensitive learning. Resampling methods achieve balance between class groups by
oversampling the group with a small size (the minority group in fairness
settings) or undersampling the group with a large size (the majority group in
fairness settings) or both. For instance, (Bao et al. 2020) carries out
classification using clustering centers in latent space to balance among the
groups, which is equivalent to undersampling all groups. Cost-sensitive
learning assigns higher weights to samples from groups with small sizes during
training such that the costs of misclassifying these samples are higher than
that from groups with large size. There are various methods on such weighting
schemes. For example, in (Huang et al. 2019), the authors balance the
representations of groups by constraining the embedding to keep inter-cluster
margins both within and between classes. Note that these works deal with class
imbalance while our work focuses on imbalance regarding sensitive attributes.

## Background

Throughout this work, we consider binary classifiers that produce estimations
$\hat{y}\in\\{0,1\\}$ for a given a dataset
$\mathcal{D}=\\{(x_{1},y_{1}),\ldots,(x_{n},y_{n})|x_{i}\in
X\subseteq\mathbb{R}^{d},y_{i}\in Y=\\{0,1\\}\\}$, where $x_{i}$ represents
vectors of attributes, and $y_{i}$ is the target label of data instance $i$.
Let the first component of $x_{i}$ describe the sensitive attribute
$s_{i}=x_{i,1}\in\\{0,1\\}$. The values of the sensitive attributes $s_{i}$
distinguish between the majority group having $n_{p}$ many samples and the
minority (i.e., sensitive or under-represented) group having $n_{u}$ many
samples. Without loss of generality, we assume that $\forall i:1\leqslant
i\leqslant n_{p}\Rightarrow s_{i}=1.\forall i:n_{p}+1\leqslant i\leqslant
n\Rightarrow s_{i}=0$.

### Fairness notions

Disparate treatment (Zafar et al. 2017) occurs when the classifier makes
different predictions on individuals from different sensitive groups when the
input features are identical. To mitigate it, the classifier should achieve
calibration across the sensitive groups: $P(\hat{y}|x,s)=P(\hat{y}|x)$.
Disparate impact (Kamiran and Calders 2016) evaluates the difference in
positive outcome rate between groups and is eliminated when the predictive
outcome $\hat{y}$ is independent of $s$: $P(\hat{y}|s=0)=P(\hat{y}|s=1)$.
Nevertheless, eliminating disparate impact does not ensure a fair classifier.
Since the sample distribution among sensitive groups is not naturally even,
the classifier might focus on the majority group while ignoring decisions on
the minority group. Furthermore, even if zero disparate impact is achieved, we
might sacrifice the classifier’s performance since statistical features of
different sensitive groups usually vary. Disparate mistreatment (Hardt, Price,
and Srebro 2016) occurs when the misclassification rates (false positives and
false negatives) of different sensitive groups are different. In this case,
the measurement of disparate mistreatment requires labeled data. Earlier
works, including (Chouldechova 2016), state that there is usually tension
among the disparate mistreatment criteria. Disparate FPR (false positive rate)
and Disparate FNR (false negative rate) are commonly used to reduce disparate
mistreatment: $P(\hat{y}\neq y|y=1,s)=P(\hat{y}\neq y|y=1)$ and $P(\hat{y}\neq
y|y=0,s)=P(\hat{y}\neq y|y=0)$.

### Wasserstein distance methods

Definition The Wasserstein distance between two distributions $\mu$ and $\nu$
is defined by
$W(\mu,\nu)=\text{min}_{\pi\in\Pi}\mathbb{E}_{(x,x^{\prime})\sim\pi}[\|x-x^{\prime}\|^{p}]$,
where $\Pi$ is the set of couplings of $\mu$ and $\nu$, i.e.,
$\Pi=\\{\pi|\int\pi(x,x^{\prime})dx^{\prime}=\mu(x)$,$\int\pi(x,x^{\prime})dx=\nu(x^{\prime})\\}$,
and $p\geqslant 1$. In the later sections of this work, $p=2$. Following the
Kantorovich-Rubinstein duality, we have the dual form of the Wasserstein
distance of $W(\mu,\nu)=\text{max}_{\|f\|_{L}\leqslant
1}\mathbb{E}_{x\sim\mu}[f(x)]-\mathbb{E}_{x^{\prime}\sim\nu}[f(x^{\prime})]$,
where the maximization is over all 1-Lipschitz functions $f$:
$\mathbb{R}^{d}\rightarrow\mathbb{R}$.

#### Fairness-aware classification

Wasserstein distance, also known as Optimal Transport (OT) distance, is a
metric in the space of measures with finite moments that can be used to
evaluate how two distributions are different from one another. A valuable
application of the properties of this metric is the computation of the
barycenter of two distributions. Such technique has been leveraged in
fairness-aware classification methods (Zehlike, Hacker, and Wiedemann 2020;
Jiang et al. 2019) to enforce statistical parity. Notably, Wasserstein Fair
Classification (WFC) (Jiang et al. 2019) quantile matches the predictions of
the sensitive group to the predictions of the barycenter of all groups.
Fairness with Continuous Optimal Transport (Chiappa and Pacchiano 2021)
introduces a stochastic-gradient fairness method based on a dual formulation
of continuous OT instead of discrete OT to improve performance.

#### Generative methods

Wasserstein GANs (WGANs) (Arjovsky, Chintala, and Bottou 2017) are based on
minimising the Wasserstein distance between a real and a generated
distribution by weight clipping to enforce a Lipschitz constraint on the
critic, improving the performance of plain GANs. WGANs with Gradient Penalty
(WGAN-GP) (Gulrajani et al. 2017) is a relaxed version of the Lipschitz
constraint, which follows that functions are 1-Lipschitz if the gradients are
of norm at most 1 everywhere.

Figure 1: Architecture of our approach. The arrows show the computational flow
for the minority (resp. majority) group in the classification task (e.g.,
predicting whether a person in the image is wearing a hat). Representation
bias is indicated by blue and red triangles. Both minority and majority groups
are mapped onto a latent space by the feature extractor. Then, majority group
instances are reweighted to match the minority group distribution, aiming to
decrease the distance with respect to the sensitive attribute.

## Our Adversarial Reweighting Approach

### Problem formulation

Consider a feature extractor $F_{\phi}:X\rightarrow Z\subseteq\mathbb{R}^{k}$
that maps raw data from the dataset $\mathcal{D}$ into a latent feature space.
The transformation function can be viewed as an embedding component for more
effective comparison of instances in latent space. A binary classifier
$C_{\theta}:Z\rightarrow Y$ with parameters $\theta$ maps the results of the
transformation $F_{\phi}(x)$ to a binary label $\hat{y}\in Y=\\{0,1\\}$. For
simplicity, we demonstrate our method in the scenario with binary sensitive
attribute. However, it is straight-forward to extend our method to handle a
multi-categorical sensitive attribute or multi-sensitive attributes (see
Appendices).

As part of our problem definition, we indicate the training objective that
minimizes the weighted empirical risk as:

$\underset{\theta}{\text{min}}\sum_{i=1}^{n}w_{i}\mathcal{L}(y_{i},(C_{\theta}\circ
F_{\phi})(x_{i})),\text{ with }w_{i}\geqslant 0$ (1)

with $\mathcal{L}$ representing cross-entropy loss. The overall pipeline we
seek to develop is illustrated in Figure 1. The feature extractor is optional
and may not be needed given low dimensionality of the data.

If all training samples receive the same weights, the classifier will tend to
focus more on the majority group leading to representation bias. We seek to
maintain the weights for samples from the minority group (i.e., $\forall
i:s_{i}=0\Rightarrow w_{i}=1$) while lowering the weights for samples from the
majority group, such that the sum of weights is the same for both groups:

$\sum_{i=1}^{n_{p}}w_{i}=n_{u}$ (2)

To avoid information loss by assigning zero weights to some samples from the
majority group, we introduce a regularization constraint to our risk
minimization term:

$\sum_{i=1}^{n_{p}}(w_{i}-\frac{n_{u}}{n_{p}})^{2}\leqslant Tn_{u}$ (3)

The sum is minimal (namely zero) if $\forall i:w_{i}=\frac{n_{u}}{n_{p}}$.
Thus, by adjusting the value of $T$ we can balance between similarity and
dissimilarity of the weights of samples from the majority group.

Together, Equations (1), (2), and (3) constrain the problem space. On their
own, however, they do not fully account for within and between sample group
differences, thus not always improving group-based fairness metrics. The
problem is now defined to seek a weighting scheme that fulfills the Equations
(1), (2), and (3) while mitigating the representation bias in a robust way.

### Adversarial reweighting

The goal of our weighting scheme is to determine weights such that the
majority and minority group weighted distributions become similar and, hence,
the classifier is less prone to biased and unfair predictions. Our adversarial
learning of data weights in the majority group targets to pay more attention
to samples in the majority group that are closer to the minority group during
the training, without completely loosing the information contained in other
samples of the majority group. We measure the similarity of weighted
distributions by the Wasserstein distance in the latent space for the reason
that approximating the Wasserstein distance in the latent space is
computationally less demanding in a low-dimensional space.

In the following, we first show that enforcing a small Wasserstein distance in
the latent space ensures small distance in the prediction space. Then, we
discuss the detailed adversarial reweighting model.

#### Theoretical proof of enforcing demographic parity

We show that enforcing the Wasserstein distance, indicated as
$W(\cdot,\cdot)$, being small in the latent space enforces it to be small in
the prediction space as well.

######  Proposition 1.

Given two measures $\mu$ and $\nu$ over a metric space $(Z,d_{Z})$ and a
$K$-Lipschitz function $C:(Z,d_{Z})\rightarrow(Y,d_{Y})$, we have that

$W(C_{\\#}\mu,C_{\\#}\nu)\leq K\cdot W(\mu,\nu)$

Where $C_{\\#}\mu$ is the push-forward measure along the function $C$. For
details of proof, please refer to Appendices.

Note that, since we deal with classifiers over a finite dataset, the
$K$-Lipschitz condition for a (binary) classifier C: $Z\rightarrow\\{0,1\\}$
amounts to asking that for every $z$ and $z^{\prime}$ such that $C(z)\neq
C(z^{\prime})$ we have that $\frac{1}{K}\leq d_{Z}(z,z^{\prime})$ because the
set $\\{0,1\\}$ is endowed with the discrete metric. Given that we consider
only finite datasets, we can always find such a $K$. Note that this result is
only valid for a given dataset and it does not generalize unless we assume
that the condition $\frac{1}{K}\leq d_{Z}(z,z^{\prime})$ for every $z$ and
$z^{\prime}$ such that $C(z)\neq C(z^{\prime})$ holds true for the new data as
well.

Thus, we have that if the Wasserstein distance is close to 0 in the latent
space, it will be close to 0 in the prediction space. Note that
$W(C_{\\#}\mu,C_{\\#}\nu)=0$ means that $C_{\\#}\mu=C_{\\#}\nu$ and this
implies demographic parity. Indeed, defining the distribution
$\zeta:=\frac{1}{2}\mu+\frac{1}{2}\nu$ describes the probability of being
sampled from either the majority or the minority groups. Then, since $\mu$ and
$\nu$ are discrete, we have that
$C_{\\#}\zeta=\frac{1}{2}C_{\\#}\mu+\frac{1}{2}C_{\\#}\nu$. Therefore,
$C_{\\#}\mu=C_{\\#}\nu$ implies that the probability of $C(z)=1$ is
irrespective of the fact that $z$ is sampled from the majority or minority
groups.

#### Adversarial reweighting model

We approximate the computation of the Wasserstein distance by a neural network
discriminator $D$ using the gradient penalty technique of WGAN-GP (Gulrajani
et al. 2017):

$W(\mu,\nu)\approx\underset{\theta_{D}}{\text{max}}\left(\mathbb{E}_{z\sim\mu}[D(z;\theta_{D})]-\mathbb{E}_{z^{\prime}\sim\nu}[D(z^{\prime};\theta_{D})]\right)$
(4)

We define the (weighted) empirical distributions of the minority group
$\mathcal{P}_{U}$ and the majority group $\mathcal{P}_{P}(\textbf{w})$ using
the Dirac delta function $\delta(\cdot)$ as:

$\displaystyle\mathcal{P}_{U}=\frac{1}{n_{u}}\sum_{i=n_{p}+1}^{n}\delta(F(x_{i})),$
(5)
$\displaystyle\mathcal{P}_{P}(\textbf{w})=\frac{1}{n_{u}}\sum_{i=1}^{n_{p}}w_{i}\delta(F(x_{i})),\text{
with }\sum_{i=1}^{n_{p}}w_{i}=n_{u}$ (6)

Then, we optimize the weights by minimizing the Wasserstein distance between
the minority and reweighted majority distributions, whereby Equations (2) and
(3) define the solution space for the weights
$\mathcal{W}=\\{\textbf{w}:\textbf{w}=(w_{1},w_{2},...,w_{n_{p}})^{T},w_{i}\geqslant
0,\sum_{i=1}^{n_{p}}w_{i}=n_{u},\sum_{i=1}^{n_{p}}(w_{i}-\frac{n_{u}}{n_{p}})^{2}\leqslant
Tn_{u}\\}$:

$\underset{\textbf{w}\in\mathcal{W}}{\text{min}}W(\mathcal{P}_{U},\mathcal{P}_{P}(\textbf{w}))$
(7)

Because of Proposition 1, we know that such minimization contributes to
reducing the disparity between majority and minority groups.

If $f$ is a measurable function and $\mu=\sum\alpha_{i}\delta(x_{i})$ a
discrete distribution, we have that
$f_{\\#}\mu=\sum\alpha_{i}\delta(f(x_{i}))$. Hence, combining Equations (4)
and (7) results into a minmax problem, yields:

$\underset{\mathbf{w}\in\mathcal{W}}{\text{min}}\underset{\theta_{D}}{\text{max}}\bigg{(}\sum_{i=1}^{n_{p}}w_{i}D(z_{i}^{p};\theta_{D})-\sum_{i=1}^{n_{u}}D(z_{i}^{u};\theta_{D})\bigg{)}$
(8)

In Equation (8), the discriminator is trained to maximize the average of its
outputs on the minority and majority group; adversarially, the weights for
samples from the majority group are learned to minimize the (reweighted)
average of the outputs of the discriminator. As a result, the samples from the
majority group with smaller discriminator outputs (closer to the minority
group) will be assigned higher weights. Therefore, defining the reweighted
cross-entropy loss on the (reweighted) data distribution in Equation (1)
mitigates the representation bias regarding the minority groups.

### Training algorithm

To train the feature extractor $F_{\phi}$ and the classifier network
$C_{\theta}$, the network parameters ($\phi$, $\theta$) and learn the weights
$w$ with $D$ are updated by fixing others. We alternately train the following
two steps.

Updating $\phi$ and $\theta$ while fixing $\mathbf{w}$ and $\theta_{D}$.
$\phi$ and $\theta$ are updated to minimize the loss in Equation (1) for $S$
steps batch-wise while $\mathbf{w}$ and $\theta_{D}$ are fixed.

Updating $\mathbf{w}$ and $\theta_{D}$ while fixing $\phi$ and $\theta$.
Embeddings of training data on both majority and minority groups are acquired
through the feature extractor $F$ while $\phi$ and $\theta$ are fixed.
$\mathbf{w}$ in Equation (8) is learned: Equation (8) is a min-max
optimization problem, the weights $\mathbf{w}$ and the parameters $\theta_{D}$
of the discriminator could be optimized alternatively. We could first fix
$w_{i}=\frac{n_{u}}{n_{p}}$ for all $i$ and optimize $\theta_{D}$ to maximize
the objective function in Equation (8) using the gradient penalty technique,
as in WGAN-GP (Gulrajani et al. 2017). Then, fixing the discriminator, we
optimize $\mathbf{w}$. We denote $d_{i}=D(F_{\theta}(x_{i});\theta_{D})$ and
$\mathbf{d}=(d_{1},d_{2},...d_{n_{p}})^{T}$. The optimization problem for
$\mathbf{w}$ becomes a constrained least squares problem:

$\underset{\mathbf{w}}{\text{min}}\,\mathbf{d}^{T}\mathbf{w},s.t.w_{i}\geqslant
0,\sum_{i=1}^{n_{p}}w_{i}=n_{u},\sum_{i=1}^{n_{p}}(w_{i}-\frac{n_{u}}{n_{p}})^{2}\leqslant
Tn_{u}$ (9)

## Experiments

We evaluate the performance of our reweighting approach on three benchmark
datasets comparing it against eight methods using four different metrics:
Accuracy, Disparate Impact, Disparate FPR and Disparate FPR.

Four fairness-agnostic methods help us to better understand issues with
unfairness. Inspired by (Chai and Wang 2022) we compare against (1) Baseline
(Neural Network (NN) based classification without fairness constraints); (2)
Simple Reweighing: NN classification with assigning same balancing weights to
samples the majority group; (3) Undersampling forms the training dataset by
balancing group sizes via undersampling from the majority group; (4)
Oversampling balances group sizes by repeating sampling from the minority
group.

We choose four further competing methods mentioned in earlier sections: (5)
Adaptive sensitive reweighting (ASR) reweights samples to balance target class
occurrences. (6) Wasserstein fair classification (WFC) matches quantiles of
the predictive distribution of the sensitive group to the all-group
Wasserstein barycenter. (7) The Fair Adversarial Discriminative (FAD) model
(Adel et al. 2019) decorrelates the sensitive information from the embeddings
by adjusting the encoding/feature extraction process using adversarial
training. (8) Fairness with Adaptive Weights (FAW) constrains the sum of
weights. They (i) are designed to address bias, (ii) follow conceptually
similar strategies, and (iii) can also be flexibly applied to different
modalities (tabular and images).

Our networks are trained on an Intel(r) Core(TM) i7-8700 CPU. The networks in
our experiments are built based on Pytorch (Paszke et al. 2019) and the
optimization in Equation (9) is performed with the python package CVXPY
(Diamond and Boyd 2016).

### Data and training details

#### Image dataset

We test three datasets based on CelebA (Liu et al. 2015) which contain 70%
male images vs. 30% female images, 80% male images vs. 20% female images and
90% male images vs. 10% female images, respectively. We use three different
distributions for the sensitive attribute to analyze in which imbalance
situation our method is more suitable. We maintain the class imbalance in the
three datasets constant, namely 70% not wearing a hat and 30% wearing a hat.
For more details on CelebA, please refer to Appendices. The classification
task is to identify whether the person in the picture is wearing a hat.

For the feature extractor $F$, we apply ResNet-18 (He et al. 2016)
architecture, pre-trained on ImageNet (Deng et al. 2009), without the last
fully-connected layer for simplicity. For the feature extractor $F_{\phi}$ and
classifier $C_{\theta}$, we use the stocastic gradient descent (SGD) algorithm
(Shamir and Zhang 2012) with a momentum of 0.9 to update $\phi$ and $\theta$.
For the discriminator $D$, we use a similar architecture as the one in
(Gulrajani et al. 2017) with three fully connected layers of 512, 256 and 128,
64, and 1 node, respectively; and without the last sigmoid function. We apply
the Adam algorithm (Kingma and Ba 2014) to update $\theta_{D}$ with a learning
rate of 0.0001. Following (Gulrajani et al. 2017), we adjust the learning rate
$\eta$ by $\eta=\frac{0.01}{(1+10p)^{-0.75}}$, where $p$ is the training
progress linearly changing from 0 to 1. We set the batch size to $n_{p}$ 700,
800, 900, and $n_{u}$ 300, 200, and 100, respectively. We update $\phi$ and
$\theta$ for 4 steps and then update $\theta_{D}$ for 1 step. Note that we
choose a relatively high batch-size because estimating the true Wasserstein
distance between distributions via batches requires relevant sample sizes.
Details on split of training and testing datasets are shown in Appendices.

Table 1: Experiment Results on CelebA

methods | simple methods | state-of-the-art methods | ours
---|---|---|---
baseline | reweighing | undersampling | oversampling | ASR | WFC | FAD | FAW | AR
Accuracy rate (%) | 95.1 (0.7) | 94.9 (0.6) | 95.3 (0.4) | 94.7 (0.9) | 93.5 (0.6) | 93.6 (0.5) | 95.0 (0.7) | 93.7 (0.6) | 94.7 (0.7)
Disparate Impact (%) | 6.0 (0.8) | 6.0 (0.4) | 5.7 (0.2) | 4.9 (0.7) | 5.1 (0.7) | 5.3 (2.4) | 5.5 (2.4) | 4.7 (0.4) | 0.8 (0.5)
Disparate FPR (%) | -29.1 (1.4) | -31.3 (9.2) | -31.8 (8.1) | -24.7 (7.5) | -26.2 (8.1) | 4.5 (1.1) | -25.7 (5.8) | -13.7 (2.1) | -17.0 (5.1)
Disparate FNR (%) | 7.3 (2.9) | 8.1 (4.2) | 7.1 (3.6) | 7.9 (1.9) | 8.2 (1.8) | 7.6 (3.9) | 6.9 (1.2) | 10.0 (1.1) | 6.5 (1.9)

(a) Experimental results of classifier (Wearing Hat) on dataset (30% female
and 70% male)

methods | simple methods | state-of-the-art methods | ours
---|---|---|---
baseline | reweighing | undersampling | oversampling | ASR | WFC | FAD | FAW | AR
Accuracy rate (%) | 95.3 (0.9) | 93.7 (0.5) | 95.0 (0.4) | 94.9 (1.6) | 93.1 (0.7) | 93.3 (0.8) | 95.0 (0.5) | 93.6 (0.8) | 95.3 (0.9)
Disparate Impact (%) | 3.7 (0.7) | 3.9 (0.9) | 3.1 (0.3) | 3.4 (0.3) | 4.3 (0.9) | 5.1 (1.4) | 18.2 (1.4) | 3.8 (0.4) | 0.7 (0.4)
Disparate FPR (%) | -16.0 (1.7) | -32.1 (7.3) | -28.9 (4.1) | -17.4 (5.0) | -2.0 (4.4) | 4.0 (1.7) | -21.0 (2.6) | -19.7 (1.1) | -22.2 (5.3)
Disparate FNR (%) | 10.3 (2.1) | 16.6 (5.4) | 11.9 (3.7) | 10.9 (3.7) | 8.1 (0.8) | -7.2 (2.1) | 10.5 (1.4) | 10.0 (1.1) | 9.0 (3.7)

(b) Experimental results of classifier (Wearing Hat) on the dataset (20%
female and 80% male)

methods | simple methods | state-of-the-art methods | ours
---|---|---|---
baseline | reweighing | undersampling | oversampling | ASR | WFC | FAD | FAW | AR
Accuracy rate (%) | 95.0 (0.5) | 93.7 (0.7) | 94.2 (0.8) | 94.1 (1.4) | 94.5 (0.5) | 92.3 (0.4) | 94.6 (0.7) | 92.5 (0.7) | 95.3 (0.2)
Disparate Impact (%) | 1.9 (0.6) | 3.3 (0.3) | 2.1 (0.3) | 1.5 (0.3) | 2.0 (0.4) | 7.9 (2.6) | 27.2 (1.0) | 1.9 (0.5) | 0.2 (0.3)
Disparate FPR (%) | -33.0 (1.3) | -27.2 (5.1) | -29.2 (1.1) | -35.5 (2.1) | -23.0 (3.4) | 17.7 (1.3) | -30.8 (1.7) | -18.7 (1.1) | -22.8 (1.0)
Disparate FNR (%) | 30.0 (1.2) | 36.6 (4.2) | 30.0 (1.3) | 29.8 (2.4) | 26.8 (0.9) | 22.2 (1.8) | 26.9 (1.8) | 26.0 (1.3) | 28.2 (2.4)

(c) Experimental results of classifier (Wearing Hat) on dataset (10% female
and 90% male)

(a) male samples assigned with the lowest weights

(b) male samples assigned with the highest weights

Figure 2: Samples from the male group with the lowest and highest weights.
Samples with the lowest weights tend to wear suits and have short hair, while
samples with the highest weights tend to have longer hair.

#### Tabular dataset

For experiments on tabular data, we use the Adult dataset (Kohavi 1996) and
the UCI German Credit Risk dataset (Dua and Graff 2017) (For more details of
the datasets, please refer to Appendices). Note that tabular datasets
generally need more preprocessing than image datasets (Borisov et al. 2022).
Note we are aware that gradient boosting would be more adequate for tabular
data, but could not find any related approach for mitigating representation
bias based on boosting. We normalize the continuous features and use one-hot
encoding to deal with the categorical features. We train the model for 50
epochs with batch-sizes of 1000 and 500 for the male and female samples. For
more details of the experiment, please refer to Appendices.

### Analysis results

#### Performance comparison

From Table 1 to Table 3, we can see that there is no sacrifice of accuracy
with our approach. At the same time, the disparate impact concerning the
sensitive attribute is mitigated, which is a crucial advantage of our
optimization over related approaches. To better understand the performance of
our method, we break down the accuracy concerning the male and female groups
and show it in Tables 4 and 5 in Appendices.

WFC did not perform so well accuracy-wise. We think it is because it adjusts
the prediction results by aligning the Wasserstein distance between the
predictions over the sensitive groups, which could reduce the accuracy rate.
While ASR is a strong competitor, it requires multiple times of training until
a convergence of the neural networks is reached, making it more expensive than
the other approaches. Similarly, this holds true for FAW. FAD fails to deal
with the imbalance problem during the decorrelation, but it maintains a high
accuracy rate.

Figure 2 shows the samples from the male groups, which are assigned the lowest
and highest weights, respectively. We see that male samples more distant from
the female distribution are down-weighted, balancing and harmonizing the male
and female distributions. Male samples closer to the female group are assigned
relatively high weights, which provides further information for the
classification task.

#### Embeddings and reweighting visualization

We visualize the learned weights of the majority group vs. the minority group
for the 70% male vs. 30% female dataset of CelebA. So we show the t-SNE
embeddings of the original and reweighted embeddings in Figure 3. On the left,
in Figure 3a, we can see that the male and female groups are not aligning
well, leading to discrimination against the female group, as described in
earlier sections. Our proposed reweighting method aligns the extracted
embeddings of the female group to that of the male group, as shown in Figure
3b, before the classification step. These visualizations, of course, only
partially explain our approach’s success in dealing with the problem of
representation bias concerning a sensitive attribute. In addition, note that
the original Wasserstein distance between the two distributions before
reweighting is 15.87, and after reweighting, the distance is 0.23. For more
details, please refer to Appendices.

(a) t-SNE before reweighting in the latent space

(b) t-SNE after reweighting in the latent space

Figure 3: t-SNE of extracted embeddings before and after reweighting of the
instances in a setting of 70 % male and 30 % female samples Table 2:
Experimental results of classifier on Adult dataset (sensitive attribute is
gender)

methods | simple methods | state-of-the-art methods | ours
---|---|---|---
baseline | reweighing | undersampling | oversampling | ASR | WFC | FAD | FAW | AR
Accuracy rate (%) | 83.1 (0.4) | 82.5 (0.3) | 82.1 (0.3) | 84.7 (0.9) | 81.6 (0.3) | 81.8 (0.5) | 82.4 (0.5) | 81.2 (0.6) | 83.0 (0.1)
Disparate Impact (%) | 17.8 (0.3) | 21.0 (0.4) | 18.7 (0.5) | 18.6 (0.4) | 0.4 (0.2) | 2.5 (1.0) | 5.7 (1.4) | 1.7 (0.4) | 1.3 (0.5)
Disparate FPR (%) | 17.0 (1.0) | 2.3 (4.7) | 9.2 (1.3) | 8.4 (1.6) | 27.2 (4.5) | -9.8 (0.7) | -8.7 (1.8) | -8.5 (2.4) | -10.5 (1.1)
Disparate FNR (%) | 6.1 (0.6) | 12.1 (3.5) | 4.2 (0.8) | 12.7 (0.7) | 2.3 (1.2) | 22.4 (1.3) | 3.2 (0.7) | 4.0 (1.7) | 7.2 (0.7)

Table 3: Experimental results of the classifier on German Credit dataset
(sensitive attribute is sex)

methods | simple methods | state-of-the-art methods | ours
---|---|---|---
baseline | reweighing | undersampling | oversampling | ASR | WFC | FAD | FAW | AR
Accuracy rate (%) | 70.1 (1.4) | 69.2 (1.3) | 65.5 (1.1) | 67.3 (2.9) | 69.1 (2.4) | 69.7 (0.5) | 68.9 (0.5) | 69.5 (0.5) | 70.0 (1.4)
Disparate Impact (%) | 15.4 (9.3) | 8.9 (3.4) | 11.9 (6.5) | 9.2 (4.7) | 6.7 (2.5) | 3.2 (1.0) | 3.5 (0.4) | 3.8 (0.5) | 1.1 (1.2)
Disparate FPR (%) | 7.5 (4.8) | 19.0 (3.7) | 13.4 (7.3) | 12.5 (8.5) | 1.5 (0.9) | 7.2 (0.8) | 1.5 (0.2) | 9.2 (2.8) | 5.3 (1.7)
Disparate FNR (%) | 6.3 (3.2) | 9.0 (3.2) | 15.7 (8.8) | 11.8 (8.7) | 0.9 (0.4) | 5.2 (2.3) | 1.1 (0.3) | 4.9 (2.5) | 6.7 (0.9)

#### Classification with noisy label

Since our method ensures the demographic parity of the predictions, it should
not be sensitive to noise labeling (possible biased labeling). We apply half
the noise corruption to the majority group and half to the minority group. We
show the performance of our method and baseline (NN-based classification
without fairness constraints) on accuracy and disparate impact under different
ratios of noise corruption in Figure 4. The disparate impact remains low when
the noise ratio changes. Moreover, again, we show no sacrifice of accuracy
when applying our method. For more details of performance on other fairness
metrics, please refer to Figure 6 in Appendices.

(a) Accuracy vs. Noise ratio

(b) Disparate Impact vs. Noise ratio

Figure 4: Change of accuracy and disparate impact under different noise ratios
on CelebA 90% male and 10% female. Note that we enforce Demographic Parity
which is selection rate parity (which our method can implicitly mitigate) and
we believe this is the reason why our method is so robust to noisy labeling.

#### Sensitivity to the choice of hyper-parameters

We have also analyzed the sensitivity of our method to the hyper-parameter $T$
mentioned earlier, in Figure 9 in Appendices, where the plots indicate that
the performance of our adversarial reweighting scheme has low sensitivity to
the choice of the hyper-parameter. In our experiments, we set $T$ at 5. For
analysis of other datasets, see Figure 10 in Appendices.

#### Ablation tests

Ablation test for MMD and JS-divergence dissimilarity measures. We also
conducted an ablation test for the Jenson-Shannon-divergence (JS) and maximum-
mean discrepancy (MMD) instead of Wasserstein distance to learn the weights in
our framework on the CelebA dataset, with 90% male and 10% female samples. In
Figures 5 and 8 (see Appendices for Figure 8), the performance of our method
using the Wasserstein distance is better than JS and MMD. Wasserstein distance
may be more suitable to measure their distance than the JS divergence when the
distributions are more disjoint. MMD with kernels may be unable to capture
very complex distances in high dimensional spaces compared to Wasserstein
distance. The Wasserstein distance is better for accuracy and disparate impact
but not necessarily better at Disparate FPR and Disparate FNR.

(a) Accuracy vs. Distance Methods

(b) Disparate Impact vs. Distance Methods

(c) Disparate FPR vs. Distance Methods

(d) Disparate FNR vs. Distance Methods

Figure 5: Sensitivity of different distance measures on CelebA dataset with
90% male and 10% female

Ablation test for assigning weights to both groups or only to the minority
group. Our method, by design, aims at reweighting the majority group to close
the gap to the Wasserstein distance between samples with different sensitivity
attribute values. Therefore, we test various reweighting schemes for both
groups in an ablation test by alternatively assigning weights to one while
treating another group with fixed weights – the Wasserstein distance between
the two groups is 0.17 for the dataset 70% male and 30% female images from
CelebA. We also test the assignment of weights to the minority group. In this
situation, the Wasserstein distance after reweighting is 1.29. As we mentioned
earlier, the Wasserstein distance of our method after reweighting is 0.23.
This suggests that the difference between reweighting both and only the
majority group is marginal, while the difference between reweighting only the
majority and the minority group is significant. This is why we reweight the
majority group in our method. For more details, please refer to Appendices.

## Discussion and Conclusion

Our work is conceptually different from previous works on fairness-aware
machine learning because it balances and harmonizes protected groups, as
defined by sensitive attributes, minimizes the empirical risk, and achieves
competitive predictive quality regarding accuracy, fairness, and robustness.
Theoretical literature explores the inherent balance between fairness and
utility, and numerous experiments have demonstrated the trade-off in practice.
Nevertheless, these trade-off discussions are often based on a fixed
distribution that does not align with our current situation. We argue that an
ideal distribution exists where fairness and utility are in harmony. Our data
reweighing combined with classifier training lets us move beyond a biased
distribution and release the trade-off. One limitation in our work lies in
that WGAN-GP might fail to approximate the Wasssertein distance correctly
(Mallasto, Montúfar, and Gerolin 2019; Stanczuk et al. 2021). Based on our
experiments, our method can mitigate the need to discard sensitive attributes
or impose specific fairness constraints, thus avoiding the issue of
determining critical hyperparameters such as regularization factors. Our
approach still permits the inclusion of further regularizations or constraints
during empirical risk minimization — though we have not yet found the need to
explore such approaches.

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##  Appendix A Appendices

### Mathematical Details

We prove that enforcing the Wasserstein distance, indicated as
$W(\cdot,\cdot)$, being small in the latent space enforces it to be small in
the prediction space as well.

###### Proof.

The proof goes as follows111The proof is almost verbatim the comment from the
user Christian Bueno about statistical divergence change under a Lipschitz
push-forward map at https://mathoverflow.net/questions/314201/how-does-a-
statistical-divergence-change-under-a-lipschitz-push-forward-map.:

$\displaystyle W(C_{\\#}\mu,C_{\\#}\nu)$ $\displaystyle=\sup_{f\in
Lip_{1}(Y)}\>\int_{Y}fdC_{\\#}\mu-\int_{Y}fdC_{\\#}\nu$ (Kantorovich duality)
$\displaystyle=\sup_{f\in Lip_{1}(Y)}\>\int_{Z}f\circ Cd\mu-\int_{Z}f\circ
Cd\nu$ (Property of the push-forward) $\displaystyle=\sup_{f\in
Lip_{1}(Y)}\>K\cdot\left(\int_{Z}\frac{f\circ C}{K}d\mu-\int_{Z}\frac{f\circ
C}{K}d\nu\right)$ $\displaystyle\leq\sup_{h\in
Lip_{1}(Z)}\>K\cdot\left(\int_{Z}hd\mu-\int_{Z}hd\nu\right)$
$\displaystyle\text{(}\frac{f\circ C}{K}\text{ is 1-Lipschitz)}$
$\displaystyle=K\cdot W(\mu,\nu)$ (4)

Where $Lip_{1}(Y)$ indicates the set of 1-Lipschitz functions
$f:Y\to\mathbb{R}$. ∎

### Dataset Details

#### CelebA

CelebA contains 202,600 face images, each endowed with 40 attributes. When we
try to construct the datasets from CelebA based on our needs, we maintain the
class imbalance in the three datasets constant, which is 70% not wearing hats
and 30% wearing hats, since class imbalance is not our priority in this paper.

#### Adult dataset

The Adult dataset was drawn from the 1994 United States Census Bureau data. It
used personal information such as education level and working hours per week
to predict whether an individual earns more or less than $50,000 per year. The
dataset is imbalanced – the instances made less than $50,000 constitute 25% of
the dataset, and the instances made more than $50,000 include 75% of the
dataset. As for gender, it is also imbalanced. We use age, years of education,
capital gain, capital loss, hours-per-week, etc., as continuous features, and
education level, gender, etc., as categorical features.

#### UCI German Credit Risk dataset

This dataset contains 1000 entries with 20 categorial/symbolic attributes. In
this dataset, each entry represents a person who takes credit from a bank.
Each person is classified as having good or bad credit risks according to
their attributes.

### Training Details and Results

#### Details of WGAN-GP adaptation for our method

In the original design of WGAN-GP of the training for one batch, the sizes of
generated and original samples are equal for the Gradient Penalty as a
regularizer to be applied. Here we need to make some changes: we control the
sum of the majority group by the weights in one batch by the Wasserstein
distance to let it be equal to the sample size of the minority group in one
batch. Then we send them for further computation of the regularizer.

#### Repetition

We repeat experiments on each dataset five times. Before each repetition, we
randomly split data into training data and test data for the computation of
the standard errors of the metrics.

#### CelebA training

For the CelebA dataset, since the original data is highly dimensional image
data, we use ResNet18 and remove the last layer as a feature extractor. The
dimension of the latent space is 512. Note that we use relatively large batch
sizes during the training, and we control the sizes of the majority and
minority constant during each batch. Papers mention that the large batch size
could cause the potential failure of the approximation using neural networks
to evaluate the distributions. Our training dataset has 10000 samples, and the
test dataset has 2000 samples for all three datasets. From Figure 6, we can
see that our method has its limitation regarding Disparate FPR and Disparate
DNR.

(a) Disparate FPR vs. Noise ratio

(b) Disparate FNR vs. Noise ratio

Figure 6: Change of disparate FPR and disparate FNR under different noise
ratios on CelebA 90% male and 10% female.

#### Convergence of the training loss

We try to show the stability of training of our method. Figure 7 shows our
method’s and baseline’s convergence for the CelebA dataset with 90% male and
10% female.

Figure 7: convergence of training loss

#### Details to approximate Wasserstein distance with and without reweighting

To approximate the Wasserstein distance before reweighting, we train the
feature extractor and discriminator without the weights assigned to any
samples. When the NNs are trained, we use the discriminator to approximate the
Wasserstein distance between the two groups.

#### Breakdown of accuracy on sensitive groups

We could see that the method sacrifices the accuracy of the majority group for
the accuracy of the minority group in Table 4 and 5.

methods | accuracy (%) |
---|---|---
male group | female group | total |
baseline | 95.2 | 94.8 | 95.1 |
our method | 94.6 | 94.9 | 94.7 |
Table 4: breakdown of accuracy on 70% male and 30% female CelebA dataset methods | accuracy (%) |
---|---|---
male group | female group | total |
baseline | 95.6 | 92.6 | 95.0 |
our method | 95.1 | 94.1 | 95.3 |
Table 5: breakdown of accuracy on 90% male and 10% female CelebA dataset

(a) Accuracy vs. Distance Methods

(b) Disparate Impact vs. Distance Methods

(c) Disparate FPR vs. Distance Methods

(d) Disparate FNR vs. Distance Methods

Figure 8: sensitivity of different distance measures on Adult dataset

(a) Accuracy vs. T

(b) Disparate Impact vs. T

(c) Disparate FPR vs. T

(d) Disparate FNR vs. T

Figure 9: Sensitivity of metrics on the change of $T$ on CelebA 90% male and
10% female.

(a) Accuracy vs. T

(b) Disparate Impact vs. T

(c) Disparate FPR vs. T

(d) Disparate FNR vs. T

Figure 10: Sensitivity of metrics to the change of $T$ on Adult dataset

#### Ablation test for assigning weights

We try to assign weights only to the minority group. We could not close the
Wasserstein distance gap and assign weights only to the majority group.
Assigning weights to both groups could achieve similar results as our method.
However, we might need an additional statistical test to claim so.

#### Training on Adult Dataset and German Credit Dataset

Since the tabular datasets have relatively lower dimensions than the image
datasets, we could avoid using feature extractors. However, we use one-hot
encoding to deal with the categorical features in the latent space. We could
have a better continuity for the Wasserstein distance approximation. From
Figure 10, we can see that the metrics are not sensitive to the change of $T$
for the Adult dataset. Figure 8 shows the sensitivity of different distance
measures on the Adult dataset.

For the feature extractor $F_{\phi}$ and the classifier $C_{\theta}$, we also
apply fully connected layers. For the discriminator $D$, we use the same
architecture in (Gulrajani et al. 2017), without the last sigmoid function. We
apply SGD algorithm with a momentum of 0.9 to update $\phi$ and $\theta$. The
learning rate of $\theta$ is ten times that of $\phi$. $\theta_{D}$ is updated
by Adam algorithm with a learning rate 0.0001. Following (Gulrajani et al.
2017), we adjust the learning rate $\eta$ of $\theta$ by
$\eta=\frac{0.01}{(1+10p)^{-0.75}}$, where $p$ is the training progress
linearly changing from 0 to 1. We update $\phi$ and $\theta$ for 2 steps then
update $\theta_{D}$ for 1 step.

#### Multi-categorical sensitive attribute situation

It is straight-forward to extend our method to handle a multi-categorical
sensitive attribute or multi-sensitive attributes by using one subgroup as
reference group and reweighing other subgroups alternatively (and in turn) to
reach a state of demographic parity. We also use Adult dataset to demonstrate
this. However, we choose $race$ here as the sensitive attribute. $race$ is
$\\{^{\prime}Amer-Indian-Eskimo^{\prime}:0,^{\prime}Asian-Pac-
Islander^{\prime}:1,^{\prime}Black^{\prime}:2,^{\prime}Other^{\prime}:3,^{\prime}White^{\prime}:4\\}$
in the dataset. We use ${}^{\prime}Asian-Pac-Islander^{\prime}$ as the
reference subgroup and reweighs samples from other subgroups. We report the
disparate impact between the subgroup ${}^{\prime}White^{\prime}$ and
${}^{\prime}Black^{\prime}$. Before and after applying our method, the
disparate impact is 15.1% and 1.7% and the accuracy is 83.1% and 82.8%.


This paper has been withdrawn by the author.