Parity-based Cumulative Fairness-aware Boosting–LABEL:lastpage

# Parity-based Cumulative Fairness-aware Boosting

Vasileios Iosifidis<EMAIL_ADDRESS>Leibniz University Hannover & L3S
Research Center, Germany.  Arjun Roy<EMAIL_ADDRESS>Freie Universität Berlin & L3S
Research Center, Germany.  Eirini Ntoutsi<EMAIL_ADDRESS>Freie
Universität Berlin & L3S Research Center, Germany.

(xxx; xxx; xxx; 2000)

###### Abstract

Data-driven AI systems can lead to discrimination on the basis of protected
attributes like gender or race. One reason for this behavior is the encoded
societal biases in the training data (e.g., females are underrepresented),
which is aggravated in the presence of unbalanced class distributions (e.g.,
“granted” is the minority class). State-of-the-art fairness-aware machine
learning approaches focus on preserving the _overall_ classification accuracy
while improving fairness. In the presence of class-imbalance, such methods may
further aggravate the problem of discrimination by denying an already
underrepresented group (e.g., females) the fundamental rights of equal social
privileges (e.g., equal credit opportunity).

To this end, we propose AdaFair, a fairness-aware boosting ensemble that
changes the data distribution at each round, taking into account not only the
class errors but also the fairness-related performance of the model defined
cumulatively based on the partial ensemble. Except for the in-training
boosting of the group discriminated over each round, AdaFair directly tackles
imbalance during the post-training phase by optimizing the number of ensemble
learners for balanced error performance (BER). AdaFair can facilitate
different parity-based fairness notions and mitigate effectively
discriminatory outcomes. Our experiments show that our approach can achieve
parity in terms of statistical parity, equal opportunity, and disparate
mistreatment while maintaining good predictive performance for all classes.

###### keywords:

fairness-aware classification; class imbalance; boosting; ensemble learning

††volume: xxx

##  1 Introduction

Increasing concerns about accountability, fairness, and transparency of AI-
based decision making systems, especially for domains of high societal impact,
have been raised over the recent years [1] as a plethora of discrimination
incidents by such systems have been reported [2, 3, 4, 5, 6]. The
discriminations are mainly against individuals or groups who share specific
characteristics like gender or race (referred to as _protected groups_
hereafter) compared to other groups (referred to as _non-protected groups_
hereafter). A growing body of research has been proposed over the recent years
to address fairness and algorithmic discrimination. These methods propose
“interventions” at the input data (the so-called, pre-processing methods),
learning algorithm (the so-called, in-processing methods), or the output model
(the so-called post-processing methods) to ensure that the model decisions are
not only correct in terms of predictive performance but also fair according to
some definition of fairness.

The vast majority of these methods, e.g., [7, 8, 9, 10, 11, 12, 13, 14, 15],
focus on optimizing for fairness while maintaining an _overall_ high
classification performance. In the case of class-imbalance though, overall
performance is not a good performance indicator for the different classes. In
the binary classification case, for example, it would ignore the performance
in the minority (also called, positive) class. Such approaches might achieve
fairness by getting an overall performance parity between the protected and
non-protected groups; however, the predictive performance of the model on the
minority class is poor as confirmed by our experiments.

Class-imbalance is an inherited problem of fairness; therefore, tackling
fairness also requires tackling imbalance [16, 17]. Our proposed approach,
AdaFair, overcomes this issue and achieves fairness while preserving good
predictive performance across all classes. AdaFair is based on AdaBoost and
extends its instance weighting strategy in each round based on the fairness of
the partial ensemble. This way, in each round, the corresponding weak learner
focuses on both complex classification examples (as in traditional boosting)
and on the discriminated group, which is dynamically identified in each
boosting round. The effect on the instance weighting is evaluated based on a
cumulative notion of fairness that considers the fairness behaviour of the
partial ensemble model until the particular round. Moreover, post-training, we
select the best sequence of weak learners, achieving high performance across
all classes and ensuring fairness. Our experiments indicate that AdaFair can
provide the best trade-off among state-of-the-art fairness-aware methods in
terms of balanced error rate and fairness.

Our contributions are summarized as follows: i) we propose AdaFair, a
fairness-aware boosting method, that achieves _parity_ between the protected
and non-protected groups (thus achieving fairness) while maintaining good
predictive performance for _all_ classes (thus tackling class-imbalance). ii)
We define the notion of cumulative fairness for the ensemble for three
different parity-based fairness notions: statistical parity, equal
opportunity, and disparate mistreatment. Based on that, we propose a dynamic
instance re-weighting schema that considers both predictive- and fairness-
performance of the model. iii) We show the superiority of our cumulative
notions of fairness vs non-cumulative alternatives in terms of model
performance and stability.

This work is an extension of our previous work [18]. The significant changes
include: i) extending AdaFair to facilitate two more parity-based notions,
namely statistical parity and equal opportunity, ii) introducing the
cumulative versions of the newly added fairness notions including the new
instance weighting schemes, iii) providing a theoretical analysis on bounding
the training error of AdaFair and iv) comparison of AdaFair to current state-
of-the-art fairness-aware methods [19, 13]. Data and source code are made
available111AdaFair (source code and data) available at:
https://iosifidisvasileios.github.io/AdaFair..

##  2 Related Work

Fairness notions. “Fair is not fair everywhere” [20], but rather the
definition of fairness depends on context. As a result, more than twenty
fairness notions exist for fairness in classification [21, 22] . One of the
earliest measures of discrimination, the so-called statistical or demographic
parity [11], measures the percentage difference in the positive predictions, ,
e.g., grant a loan application, between the protected and non-protected
groups. However, this definition only requires a balanced representation of
both groups in the positive class ignoring whether the selected instances are
qualified or not [23]. A more recent measure, called equal opportunity [12,
21], alleviates this pitfall by measuring the percentage difference between
true positive rates for both groups. Disparate mistreatment [8] extends equal
opportunity by considering the difference of correctly classified instances
between protected and non-protected groups for all classes.

Pre-processing approaches. One of the most common causes of ML discrimination
arises from biases in historical data. Pre-processing methods aim to deal with
this issue by changing the underlying data to restore balance between the
protected and non-protected groups. Methods falling under this category are
typically model-agnostic; therefore, any classifier is applicable after pre-
processing. Example methods include massaging [14] that changes the class-
labels of carefully selected instances, re-weighting [9] that assigns
different weights to the instances of the protected group, uniform- or
preferential-sampling [11] and data augmentation [24] that enriches the
minority group through pseudo-instances. Another interesting line of works by
[10, 25], perform data transformations to eliminate existing correlations
between class labels and protected attributes. Some of these approaches have
been also extended to non-stationary data [26].

In-processing approaches. These approaches aim to mitigate discrimination
during model training by extending the objective function of the learner to
account for discrimination using regularization or constraints. In [15, 27],
for example, the entropy-based splitting criterion for decision tree induction
is modified based on statistical parity to also consider the fairness of the
splitting decisions. In [28] a regularization approach is proposed that scales
down the correlation (mutual information) of the sensitive features and the
class attribute to avoid outcomes based on these features. [23] introduces the
notion of “individual fairness-constraints” that impose similar treatment to
similar instances. In [8], a set of constraints which minimize disparate
mistreatment, is added to a logistic regression model. Finally, [7] assume
that the ground truth is biased and propose to estimate the true unbiased
labels by iteratively re-adjusting the instance weights while minimizing
disparate mistreatment.

Post-processing approaches. Post-processing approaches can be divided into two
subcategories: the ones that change the decision boundary of the model (white-
box approaches) and the ones that directly change the predictions of the model
(black-box approaches). In the first category, belongs [13] which shifts the
decision boundary of AdaBoost to minimize statistical parity. An extension of
this idea has been applied for non-stationary data [29, 30]: the decision
boundary is tweaked online to tackle concept drifts and to mitigate
discriminatory outcomes as evaluated by statistical parity or equal
opportunity. In [31] the authors alter the confidence of CPAR classification
rules and in [32] the authors change Naïve Bayes probabilities considering
fairness. Black-box approaches have no access to the learner but only to its
decisions. For example, [12] set thresholds to the model predictions to
achieve the same error rates for protected and non-protected groups. An
extension of this work [33] analyzes how to obtain calibrated classifiers with
the same error rates among different groups.

##  3 Basic concepts and definitions

We assume a dataset $D$ of $N$ i.i.d. samples drawn from the joint
distribution $P(F,S,y)$: $S$ denotes the set of _protected attributes_ such as
gender or race, $F$ denotes other _non-protected attributes_ and $y$ is the
class label. For simplicity, we consider that the classification problem is
binary, that is, $y\in\\{+,-\\}$ and that there exists a single protected
attribute $S$, also binary: $S\in\\{s,\bar{s}\\}$ with $s$ and $\bar{s}$
denoting the _protected_ and _non-protected group_ , respectively. We use the
notation $s_{+}$ ($s_{-}$), $\bar{s}_{+}$ ($\bar{s}_{-}$) to denote the
protected and non-protected groups for the positive (negative, respectively)
class.

Fairness measures: As already discussed in Section 2, we adopt three different
parity-based fairness measures: _St atistical Parity_ (shortly $St.Parity$),
_Eq ual Opportunity_ (shortly $Eq.Op.$), and _D isparate _M_ istreatment_
(shortly $D.M.$).

$St.Parity$ measures the difference in the positive prediction rates between
the protected and non-protected groups and is defined as:

$St.Parity=P(\hat{y}=+|\bar{s})-P(\hat{y}=+|s)$ (1)

where $\hat{y}$ denotes the prediction. $St.Parity$ takes values in the [0-1]
range with 0 standing for no discrimination and 1 for maximum/worse
discrimination.

$Eq.Op.$ measures the difference of the false negative prediction rates
between the protected and non-protected groups. It extends $St.Parity$ by
considering not only the predictions but also the ground truth labels.
Formally:

$Eq.Op.=P(y\neq\hat{y}|\bar{s}_{+})-P(y\neq\hat{y}|s_{+})=\delta FNR$ (2)

We denote by $\delta FNR$ the difference in false negative rates between the
protected and non-protected groups. $Eq.Op.$ also takes values in the [0-1]
range.

Finally, $D.M.$ measures the difference in prediction errors between the
protected and non-protected groups. It extends $Eq.Op.$ by considering not
only the positive misclassifications but also the negative misclassifications.
Similar to $\delta FNR$ above, we denote by $\delta FPR$ the difference in
false positive rates between the protected and non-protected groups, namely:

$\displaystyle\delta FPR=P(y\neq\hat{y}|\bar{s}_{-})-P(y\neq\hat{y}|s_{-})$
(3)

Then, $D.M.$ is defined as follows:

$D.M.=|\delta FPR|+|\delta FNR|$ (4)

$D.M.$ values lie in the [0-2] range as $|\delta FPR|$, $|\delta FNR|$ lie in
the [0-1] range.

The goal of fairness-aware classification is to find a mapping
$f:(F,S)\rightarrow y$ that achieves good predictive- and fairness-performance
(in our case, the latter is evaluated in terms of $St.Parity$, $Eq.Op.$, or
$D.M.$).

Predictive performance: In the context of fairness-aware learning, predictive
performance is typically assessed via _error rate_ , e.g., [7, 8, 9, 10, 11,
12, 13, 14, 15], defined as:

$ER=\frac{FN+FP}{TP+TN+FN+FP}$ (5)

where $FP,FN,TP,TN$ are the false positive, false negative, true positive,
true negative cases, respectively. However, optimizing for the error rate is
problematic in cases of _class imbalance_. A possible outcome in such a case
is that the classifier will misclassify most (in the extreme case all) of the
minority instances while correctly classifying the majority. In this scenario,
the error rate ($ER$) will still be low despite the poor performance in the
minority class, and w.r.t fairness, a classifier might still be fair, e.g.,
$D.M.\approx 0$, as the difference between the FPRs, FNRs for each group will
be low (c.f., Equations (3),(4)). However, in such a case, the non-
discriminatory behavior of the classifier would be achieved by drastically
reducing the correct predictions for the minority class, with the extreme case
of misclassifying all minority instances. As we will see in the experiments
section, many of the datasets in this domain exhibit high class imbalance
(c.f., Table 1) and therefore, tackling fairness requires also tackling
imbalance.

Our goal in this work is to minimize discriminatory outcomes (as measured by
$St.Parity$, $Eq.Op.$, or $D.M.$) while maintaining good predictive
performance for _both_ classes. To this end, we propose (c.f., Section 4.2.2)
to replace the error rate (which is not a good performance indicator in case
of class-imbalance, as we also show experimentally in Section 5) with the
_balanced error rate (BER)_ which is the average of the errors on each class
[34]:

$BER=1-\frac{1}{2}\cdot(\frac{TP}{TP+FN}+\frac{TN}{TN+FP})=1-\frac{1}{2}\cdot(TPR+TNR)$
(6)

where $TPR$ ($TNR$) is the true positive rate (true negative rate,
respectively).

AdaBoost: Our classification model is based on AdaBoost [35], an ensemble
technique that combines multiple weak learners to create a strong learner. The
weak learners are trained sequentially, each trying to correct the errors of
its predecessor by adjusting the instance weights accordingly. AdaBoost takes
as input the number of boosting rounds $T$ and trains an ensemble of weak
learners $h_{[1-T]}$, by training each weak learner $h_{j+1},j:1-T$ with the
updated weight distribution from the previous round $j$. The weight
distribution is updated based on the errors of the previous weak learner as
follows:

$D_{i}^{j+1}\leftarrow\frac{1}{Z_{j}}D_{i}^{j}\cdot\exp{(-\alpha_{j}\cdot
y_{i}\cdot h_{j}(x_{i}))}$ (7)

where $D_{i}^{j}$ is the weight of instance $i$ in the current boosting round
$j$, $Z_{j}$ is the normalization factor and $\alpha_{j}$ is the weight of the
weak learner $h_{j}$ defined as: $\alpha_{j}=\frac{1-err_{j}}{err_{j}}$ (where
$err_{j}$ is the error rate of $h_{j}$ on the training set).

The final ensemble uses a weighted majority schema at the prediction phase,
i.e., for an instance $x$ the prediction is derived as:
$H(x)=\sum_{j=1}^{T}\alpha_{j}h_{j}(x)$.

##  4 Parity-Based Fairness-Aware Cumulative Boosting

AdaBoost and boosting in general divide the complex learning problem into
lower complexity sub-problems and then combine their solutions (sub-models)
into an overall (global) model. Intuitively, such a technique is highly
promising for fairness-aware learning, as it is easier to tackle the fairness
problem in the simpler sub-models than in the complex global model. However,
adopting AdaBoost for fairness requires careful interventions in the data
distribution that take into account both predictive and fairness-related
performance (Section 4).

We tailor AdaBoost to fairness by adjusting the re-weighting process, which
traditionally focuses on the misclassifications of the previous weak learner
$h_{j}$ for training the next weak learner $h_{j+1}$ (c.f., Equation (7)). In
particular, we directly consider the fairness behaviour of the model in the
weighting process by introducing fairness-related costs. Moreover, for the
fairness-related costs, we don’t rely only on the fairness behavior of the
previous single weak learner $h_{j}$, but on the fairness behavior of the
_partial ensemble_ $H_{1:j}$. By taking into account the “history” of the weak
learners for fairness-related interventions, we aim to achieve smoother
interventions based on the _cumulative_ performance of the model rather than
on the varying performances of individual weak learners.

We first introduce the cumulative fairness costs based on the adopted fairness
notions and their corresponding cumulative versions, namely cumulative
$St.Parity$ ( Section 4.1.1), cumulative $Eq.Op.$ (Section 4.1.2) and
cumulative $D.M.$ (Section 4.1.3). The fairness-aware interventions in the
distribution re-weighting process are described in Section 4.2.1. A
theoretical analysis of the training error is provided in Section 4.3.
Finally, we optimize the number of weak learners in the final ensemble based
on the balanced error rate and thus directly considering class imbalance in
the best model selection (Section 4.2.2).

###  4.1 Cumulative fairness notions and fairness costs

Let $j\in[1,T]$ be the current boosting round and
$H_{1:j}=\\{h_{1},\cdots,h_{j}\\}$ the sequence of weak learners up to $j$,
i.e., the partial ensemble. For an instance $x$, the partial ensemble decides
according to: $H_{1:j}(x)=sign(\sum_{i=1}^{j}\alpha_{i}h_{i}(x))$. We define
the fairness-related costs based on the cumulative behavior of the model,
i.e., based on the partial ensemble $H_{1:j}$. In the following subsections,
for each fairness notion ($St.Parity$, $Eq.Op.$, $D.M.$), we first define
their cumulative counterparts and then the fairness costs.

####  4.1.1 Cumulative Statistical Parity

The _cumulative statistical parity_ in round $j$, denoted by $\delta
SP_{1:j}$, evaluates the parity in the positive predictions of the partial
ensemble $H_{1:j}$ between the protected and non-protected groups. Formally:

$\displaystyle\delta
SP_{1:j}=\frac{\sum\limits_{i,x_{i}\in\bar{s}}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})=+\right]}{|\bar{s}|}-\frac{\sum\limits_{i,x_{i}\in
s}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})=+\right]}{|s|}$
(8)

where the function $\mathbb{I}(\cdot)$ returns 1 iff the expression within is
true, otherwise 0.

If there is no parity, i.e., $\delta SP\neq 0$, we change the weights of the
training instances so that the discriminated group is boosted extra in the
next round $j+1$. Note that vanilla AdaBoost already boosts the misclassified
instances for the next round. Our weighting, therefore, aims at achieving
parity between the protected and non-protected groups. To this end, we assign
fairness-related costs to the discriminated group. More formally, the
fairness-related cost $u_{i}^{j}$, for an instance $x_{i}$ in the boosting
round $j$ is computed as follows:

$\displaystyle u_{i}^{j}=\begin{cases}|\delta
SP_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
SP_{1:j}|>\epsilon),x_{i}\in s,sign(\delta SP_{1:j})=+\\\ |\delta
SP_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
SP_{1:j}|>\epsilon),x_{i}\in\bar{s},sign(\delta SP_{1:j})=-\\\ 0,&otherwise\\\
\end{cases}$ (9)

where $u_{i}^{j}\in[0,1]$, $sign()$ is the sign function, and parameter
$\epsilon\in\mathbf{R^{+}}$ reflects the tolerance to unfairness and is
typically set to zero or to a very small value222The notions $u_{i}^{j}$ and
$\epsilon$ will bear the same meaning for the rest of the section.. The signs
(+/-) of $\delta SP_{1:j}$ denote which group is discriminated and should be
boosted w.r.t. fairness, while $\epsilon$ is a condition for the necessity of
fairness-related costs in the upcoming round $j+1$. For example, if in round
$j$ the group $s$ is discriminated, which means $\delta SP_{1:j}>\epsilon$,
then misclassified instances $x_{i}$ in this group will receive fairness-
related costs for the next round. Note that all misclassified instances of the
discriminated group will receive the same cost in the boosting round $j$.
However, the costs are dynamically estimated in each round.

####  4.1.2 Cumulative Equal Opportunity

The _cumulative equal opportunity_ in round $j$ evaluates the parity in the
false negative prediction rates of the partial ensemble $H_{1:j}$ between the
protected and non-protected groups. Formally:

$\displaystyle\delta
FNR_{1:j}=\frac{\sum\limits_{i,x_{i}\in\bar{s}_{+}}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})\neq
y_{i}\right]}{|\bar{s}_{+}|}-\frac{\sum\limits_{i,x_{i}\in
s_{+}}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})\neq
y_{i}\right]}{|s_{+}|}$ (10)

Similar to cumulative statistical parity, cumulative equal opportunity assigns
fairness-related costs in each round to instances that belong to an unfairly
treated group. For an instance $x_{i}$ in the boosting round $j$, the
fairness-related cost $u_{i}^{j}$ is computed as follows:

$\displaystyle u_{i}^{j}=\begin{cases}|\delta
FNR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FNR_{1:j}|>\epsilon),x_{i}\in s_{+},sign(\delta FNR_{1:j})=+\\\ |\delta
FNR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FNR_{1:j}|>\epsilon),x_{i}\in\bar{s}_{+},sign(\delta FNR_{1:j})=-\\\
0,&otherwise\\\ \end{cases}$ (11)

For example, if the group $s_{+}$ is discriminated in round $j$, which means
$\delta FNR_{1:j}>\epsilon$, then the misclassified instances in this group
will be boosted in the next round based on Equation 11.

####  4.1.3 Cumulative Disparate Mistreatment

_Cumulative disparate mistreatment_ extends cumulative equal opportunity by
considering parity among protected and non-protected groups for both the
positive and negative classes. We define it in terms of $\delta FPR$, $\delta
FNR$ of the partial ensemble $H_{1:j}$. Similar to $\delta FNR_{1:j}$ (c.f.,
Equation (10)), $\delta FPR_{1:j}$ is defined as:

$\displaystyle\delta
FPR_{1:j}=\frac{\sum\limits_{i,x_{i}\in\bar{s}_{-}}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})\neq
y_{i}\right]}{|\bar{s}_{-}|}-\frac{\sum\limits_{i,x_{i}\in
s_{-}}1\cdot\mathbb{I}\left[\sum\limits_{k=1}^{j}\alpha_{k}h_{k}(x_{i})\neq
y_{i}\right]}{|s_{-}|}$ (12)

In the boosting round j, miss-classified instances $x_{i}$ of the
discriminated group are boosted extra based on fairness-related costs
$u_{i}^{j}$ defined as:

$\displaystyle u_{i}^{j}=\begin{cases}|\delta
FNR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FNR_{1:j}|>\epsilon),x_{i}\in s_{+},sign(\delta FNR_{1:j})=+\\\ |\delta
FNR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FNR_{1:j}|>\epsilon),x_{i}\in\bar{s}_{+},sign(\delta FNR_{1:j})=-\\\ |\delta
FPR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FPR_{1:j}|>\epsilon),x_{i}\in s_{-},sign(\delta FPR_{1:j})=+\\\ |\delta
FPR_{1:j}|,&if~{}\mathbb{I}((y_{i}\neq h_{j}(x_{i}))\land|\delta
FPR_{1:j}|>\epsilon),x_{i}\in\bar{s}_{-},sign(\delta FPR_{1:j})=-\\\
0,&otherwise\\\ \end{cases}$ (13)

Again, only misclassified instances are susceptible to receive the fairness-
related cost (the order of the cost assignments inside the equation does not
matter). Similar to the other measures, all miss-classified instances of the
discriminated group will receive the same cost in a given boosting round $j$.
Still, the costs might change across the rounds as they depend on the partial
ensemble.

###  4.2 The AdaFair Algorithm

AdaFair is a sequential ensemble that extends Adaboost for fairness-aware
learning under class-imbalance. The algorithm consists of two steps. The first
step (Section 4.2.1) is the in-training phase of the ensemble based on the
selected cumulative fairness notion and its fairness-related costs (Section
4.1). The second step (Section 4.2.2) is the post-processing phase, in which
the algorithm selects the optimal partial ensemble that offers the minimum
weighted summed loss of (balanced) predictive performance and fairness.

####  4.2.1 In-processing Distribution Update

The main difference to vanilla AdaBoost is the weight distribution update
formula (c.f., Equation (7)) which now also considers the fairness-related
costs $u_{i}^{j}$. In particular, the data distribution is updated as follows:

$D^{j+1}(i)=\frac{D^{j}(i)C^{j}_{i}\exp{(-\alpha_{j}y_{i}h_{j}(x_{i})})}{Z_{j}}$
(14)

For convenience, we use $C_{i}^{j}=(1+u_{i}^{j})$ instead of $u_{i}^{j}$. For
example, if $u_{i}^{j}=0$ then $C_{i}^{j}=1$ and the instance is not affected.
In addition, $u_{i}^{j}\in[0,1]$; therefore, the fairness-related cost would
degrade the instance’s weight instead of boosting it.

The normalization factor $Z_{j}$ ensures $D^{j+1}$ is a probability
distribution:

$\displaystyle
Z_{j}=\sum\limits_{i=1}^{N}D^{j}(i)C^{j}_{i}\exp{(-\alpha_{j}y_{i}h_{j}(x_{i}))}$
(15)

The ensemble training is shown in Algorithm 1. Instance weights $D_{i}^{1}$
and fairness-related costs $u_{i}^{1}$ are initialized (line 1). In each
boosting round $j:1-T$ (lines 2–12), a weak learner $h_{j}$ is trained upon
the current weight distribution $D^{j}$ (line instance 3) and the
$\alpha_{j}$, and fairness-related costs $u_{i}^{j}$ are computed (lines 4,
and 5, respectively). The new instance weights are estimated (line 8).

After the in-training learning phase, AdaFair selects, post-training, the best
sequence of weak learners (line 13, Algorithm 1), which achieves the best
trade-off between balanced and standard error rate as specified by a user-
defined parameter $c$ (Equation (16)). The post-training phase directly
tackles imbalance and is discussed hereafter (Section 4.2.2).

Input: $D=(x_{i},y_{i})_{1}^{N},T,\epsilon,c$

Result: Ensemble $H$

1 Initialize $D_{i}^{1}=1/N$, $C_{i}^{1}=1$, and $u_{i}^{1}=0$, for
$i=1,2,\dots,N$;

2 for  _j=1 to $T$_ do

3 Train a classifier $h_{j}$ to the training data using weights $D^{j}$;

4 Compute the weight $\alpha_{j}$ (Equation (22));

5 Compute fairness-related costs $u_{i}^{j}$ based on a given fairness notion
(Equations (8), (10), (12));

6 $C_{i}^{j}=(1+u_{i}^{j})$;

7

8 Update the distribution as:

9 $D_{i}^{j+1}\leftarrow\frac{1}{Z_{j}}D_{i}^{j}\cdot
C_{i}^{j}\cdot\exp{(-\alpha_{j}\cdot y_{i}\cdot h_{j}(x_{i}))}$

10 // $Z_{j}$ is normalization factor;

11 if  _Termination condition does not hold (Equation ( 23))_ then

12 break;

13

14

15 end for

Return best weak learner sequence (Equation (16)) using parameter $c$

Algorithm 1 AdaFair algorithm

####  4.2.2 Post-processing Model Selection based on Balanced Performance

The number of weak learners $T$ is provided as input to AdaFair, similarly to
AdaBoost. We propose to refine the model by finding the best (sub)sequence of
weak learners $1\cdots\theta,\theta\leq T$ that achieves good performance for
both classes and is fair according to the chosen fairness measure. To this
end, we propose to optimize for the balanced error rate $BER$ (Equation (6))
instead of the standard error rate $ER$ (Equation (5)). In case of balanced
data, BER corresponds to ER. To allow for different combinations of ER and BER
in the $\theta$ computation, we consider both ER and BER in the objective
function as follows:

$\operatorname*{\arg\min}_{\theta}~{}(c\cdot BER_{\theta}+(1-c)\cdot
ER_{\theta}+F.M._{\theta})$ (16)

where $F.M.$ can be one of the aforementioned cumulative fairness measures
(Equations (8), (10), (12)). The parameter $c$ controls the impact of BER and
ER in the computation. The selection of $\theta$ is performed based on a
validation set (more details on the validation set in Section 5.1.3). A
detailed evaluation of parameter’s $c$ impact in the performance of AdaFair is
presented in Section 5.7.

Our approach _directly_ tackles class-imbalance in the post-processing phase
by selecting the best sequence of weak learners according to Equation (16). We
have also investigated the in-training mitigation of class imbalance. In
particular, we transformed AdaFair into a cost-sensitive learner by inserting
misclassification costs for each class. However, in our preliminary
investigations, the interplay between fairness-related costs and class-related
miss-classification costs resulted in an unstable model. We still believe that
such an approach is promising and we plan to pursue this direction in our
future work.

###  4.3 Bounding the training error

The update of the weight distribution of AdaFair is given in Equation (14).
Following the same reasoning as in [36], by unravelling Equation (14), we
obtain:

$\displaystyle D^{t+1}(i)$
$\displaystyle=D^{1}(i)\times\frac{C^{1}_{i}\exp{\left(-\alpha_{1}y_{i}h_{1}(x_{i})\right)}}{Z_{1}}\times\cdots\times\frac{C^{t}_{i}\exp{\left(-\alpha_{t}y_{i}h_{t}(x_{i})\right)}}{Z_{t}}$
(17)
$\displaystyle=\frac{D^{1}(i)\prod\limits_{j=1}^{t}C^{j}_{i}\exp{(-\sum\limits_{j=1}^{t}\alpha_{j}y_{i}h_{j}(x_{i}))}}{\prod\limits_{j=1}^{t}Z_{j}}$

From Equation (17), we get:

$\displaystyle
D^{1}(i)\exp{(-\sum\limits_{j=1}^{t}\alpha_{j}y_{i}h_{j}(x_{i}))}=\frac{D^{t+1}(i)}{\prod\limits_{j=1}^{t}C^{j}_{i}}(\prod\limits_{j=1}^{t}Z_{j})$
(18)
$\displaystyle\implies\frac{1}{N}\exp{(-\sum\limits_{j=1}^{t}\alpha_{j}y_{i}h_{j}(x_{i}))}=\frac{D^{t+1}(i)}{\prod\limits_{j=1}^{t}C^{j}_{i}}(\prod\limits_{j=1}^{t}Z_{j})~{}~{}~{}~{}[\because\forall
i,D^{1}(i)=\frac{1}{N}]$

Then, the training error of the final classifier $H$ is bounded as:

$\displaystyle Pr[H(x_{i})\neq y_{i}]=\frac{1}{N}\sum\limits_{H(x_{i})\neq
y_{i}}{\mathbf{1}}\leq\sum\limits_{i=1}^{N}\frac{1}{N}\exp{(-\sum\limits_{j=1}^{t}\alpha_{j}y_{i}h_{j}(x_{i}))}$
(19) $\displaystyle\implies Pr[H(x_{i})\neq
y_{i}]\leq\sum\limits_{i=1}^{N}\frac{D^{t}(i)}{\prod\limits_{j=1}^{t}C^{j}_{i}}(\prod\limits_{j=1}^{t}Z_{j})$

There exists a constant $\gamma$, such that $\forall
i,\gamma<\prod\limits_{j=1}^{t}C^{j}_{i}$. Then,

$\displaystyle Pr[H(x_{i})\neq
y_{i}]\leq(\prod\limits_{j=1}^{t}Z_{j})\sum\limits_{i=1}^{N}\frac{D^{t}(i)}{\gamma}$
(20) $\displaystyle\implies Pr[H(x_{i})\neq
y_{i}]\leq\frac{1}{\gamma}\prod\limits_{j=1}^{t}Z_{j}~{}~{}~{}~{}~{}[\because\sum\limits_{i=1}^{N}D^{t}(i)=1]$

Since $\gamma$ is a constant, in order to minimize the training error
(Equation (20)), parameter $Z$ needs to be minimized on each boosting round.
Therefore, the objective in each boosting round $t$ is to find $\alpha_{t}$
that minimizes $Z_{t}$. According to [37], once
$y_{i}h_{t}(x_{i})\in\\{-1,1\\}$ holds, the choice of $\alpha_{t}$ for each
$h_{t}$ can be obtained with the help of the following approximation:

$\begin{split}&Z_{t}=\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\exp{\left(-\alpha_{t}y_{i}h(x_{i})\right)}\\\
&\leq\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\left(\frac{1-y_{i}h_{t}(x_{i})}{2}e^{\alpha_{t}}+\frac{1+y_{i}h_{t}(x_{i})}{2}e^{-\alpha_{t}}\right)\\\
\end{split}$ (21)

To estimate the $\alpha_{t}$ that minimizes $Z_{t}$, we need to solve for
$\alpha_{t}$ that minimizes the approximation upper bound in Equation (21).

$\begin{split}&\displaystyle{\frac{\partial}{\partial\alpha_{t}}\left(\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\left(\frac{1-y_{i}h_{t}(x_{i})}{2}e^{\alpha_{t}}\right)+\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\left(\frac{1+y_{i}h_{t}(x_{i})}{2}e^{-\alpha_{t}}\right)\right)=0\Rightarrow}\\\
&e^{\alpha_{t}}\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\left(\frac{1-y_{i}h_{t}(x_{i})}{2}\right)=e^{-\alpha_{t}}\sum\limits_{i=1}^{N}D^{t}(i)C^{t}_{i}\left(\frac{1+y_{i}h_{t}(x_{i})}{2}\right)\Rightarrow\\\
&\alpha_{t}=\frac{1}{2}\log\left(\frac{\sum\limits_{i,y_{i}=h_{t}(x_{i})}^{N}C_{i}^{t}D^{t}(i)}{\sum\limits_{i,y_{i}\neq
h_{t}(x_{i})}^{N}C_{i}^{t}D^{t}(i)}\right)\end{split}$ (22)

To preserve the property of AdaBoost, i.e., to ensure $\alpha_{t}$ is strictly
positive, the following must hold:

$\sum\limits_{i,y_{i}=h(x_{i})}C^{t}_{i}D^{t}(i)>\sum\limits_{i,y_{i}\neq
h(x_{i})}C^{t}_{i}D^{t}(i)$ (23)

##  5 Evaluation

We evaluate the predictive performance and fairness behavior of AdaFair vs
other related approaches (Sections 5.2, 5.3 and 5.4). Regarding predictive
performance, we report on both accuracy (Equation (5)) and balanced accuracy
(Equation (6)), whereas for fairness we report on statistical parity (S.P.),
equal opportunity (Eq.Op.) and disparate mistreatment (D.M.) (c.f., Section
3).

Another goal of our experiments is to understand the behaviour of AdaFair. To
this end, we investigate the effect of cumulative vs non-cumulative fairness
(Section 5.6) and the impact of adopting balanced error rate vs error rate
(Section 5.7) for the post-processing model selection. We provide the details
on the datasets, baselines, parameter selection and evaluation in Section 5.1.

###  5.1 Experimental setup

####  5.1.1 Datasets

We evaluate our approach on four real-world datasets whose characteristics are
summarized in Table 1. They comprise a suitable benchmark due to their diverse
characteristics, namely cardinality, dimensionality and class imbalance.

Adult census [38] Bank [38] Compass [6] KDD census [38] #Instances 45,175
40,004 5,278 299,285 #Attributes 14 16 9 41 Sen.Attr. Gender Marit. Status
Gender Gender Prot.Group (s) Female Married Female Female Class ratio
($+$:$-$) 1:3.03 1:7.57 1:1.12 1:15.11 Positive class >50K subscription
recidivism >50K

Table 1: An overview of the datasets used in our experiments.

Adult census [38] dataset contains demographic data from the U.S. The task is
to predict whether the annual income of a person will exceed 50K dollars. The
protected attribute is $S=Gender$ with $s=female$ being the protected group;
the positive class is people receiving more than 50K. We remove duplicate
instances and instances containing missing values. The positive to negative
class ratio is 1:3 (exact ratio 24%:76%).

Bank dataset [38] is related to direct marketing campaigns of a Portuguese
banking institution. The task is to determine if a person will subscribe to
the product (bank term deposit). As positive class we consider people who
subscribed to a term deposit. We consider as $S=marital~{}status$ with
$s=married$ being the protected group. The dataset suffers from severe class
imbalance, with a positive to negative ratio of 1:8 (exact ratio 11%:89%).

Compass dataset [6] stores record about prisoners in Broward County. The task
is to predict (recidivism), namely if a person will be re-arrested within two
years. We consider recidivism as the positive class and $S=Gender$ with
$s=female$ as the protected group. For this dataset, we followed the pre-
processing steps of [8]. The dataset with a positive to negative ratio of
$46\%:54\%$ is almost balanced.

KDD census [38] has the same prediction task as the Adult census dataset.
However, in KDD census “the class labels were drawn from the total person
income field rather than the adjusted gross income” [38].

####  5.1.2 Baselines

We evaluate AdaFair against state-of-the-art methods for each fairness
measure.

For _statistical parity_ , we employ AdaBoost SDB [13], which trains a vanilla
AdaBoost, and afterwards, it tweaks the decision boundary of the induced model
(based on a validation set) to mitigate discriminatory outcomes.

For _equal opportunity_ , we employ FAE [19], an ensemble that combines pre-
and post-processing steps to mitigate unfair outcomes and to tackle class
imbalance. It pre-processes the data by sampling the dataset to generate equi-
sized samples w.r.t the protected attribute and assigns them to a set of bags
used to induce an ensemble of AdaBoost models. In the post-processing phase,
it shifts the decision boundary of the ensemble to account for fairness.

For _disparate mistreatment_ , we employ the methods by Zafar et al. [8] and
Krasanakis et al. [7]. Zafar et al. [8] presents the fairness problem as a set
of convex-concave constraints to minimize discriminatory outcomes and solve it
using a logistic regression model. Krasanakis et al. [7] assume the existence
of latent fair classes and propose an iterative training approach towards
those classes by altering the instance weights. We have selected these methods
as they follow a different line of reasoning, constraints [8] vs hidden
unbiased labels [7].

In addition, we compare against two _fairness-agnostic_ boosting methods:
vanilla AdaBoost [35] and SMOTEBoost [39]. SMOTEBoost is an extension of
AdaBoost for imbalanced data which tackles imbalance by generating in each
boosting round new synthetic instances of the minority class using SMOTE [40].
The goal of employing SMOTEBoost is to see whether the fairness problem can be
addressed by only tackling class imbalance.

Finally, to study the behaviour of AdaFair, we also compare it against a
variation AdaFair NoCumul that computes the fairness-related costs in each
round based on only the fairness evaluation of the current weak learner
$h_{j}$, instead of the partial ensemble $H_{1:j}$. The goal of this baseline
is to clarify the understanding of the impact of the cumulative fairness
notion (Section 5.6).

####  5.1.3 Parameter selection and evaluation

We follow the evaluation setup as in [8, 7] by splitting each dataset randomly
into train $(50\%)$ and test set $(50\%)$ and report on the average of 10
random splits. We set $\epsilon=0$ as a threshold for all fairness measures,
which means zero tolerance to discrimination. For extracting the validation
set (which is used for estimating $\theta$, Section 4.2.2) we perform a
stratified split (67% training and 33% validation set). Our method is
instantiated in each boosting rounds with decision trees of depth 1 (decision
stumps) as weak learners. For the total number of boosting rounds, we set
$T=200$ (same for the other ensemble approaches: AdaBoost, AdaBoost SDB, FAE,
and SMOTEBoost, c.f., Section 5.1.2). We analyze the effect of $T$ in Section
5.5. For Krasanakis et al. and Zafar et al. methods, we employ their default
(suggested) parameters. For SMOTEBoost, we set $N$ (the number of synthetic
instances generated per round) to 2, 100, 100 and 500 for datasets Compass,
Adult census, Bank and KDD census, respectively. Furthermore, for experiments
in Sections 5.4, 5.5 and 5.6, we set parameter $c=1$ (c.f., Equation (16)),
that is the proposed AdaFair optimized for balanced error rate; the effect of
$c$ is studied in Section 5.7.

###  5.2 Statistical Parity: Predictive and fairness performance

In Figure 1 we report on predictive performance (both accuracy and balanced
accuracy, Bal.Acc. for short) and on fairness-related performance, namely
statistical parity (St. Parity, for short). We also report on the TPR and TNR
for both protected and non-protected groups to showcase the fairness and
accuracy of the approaches for both groups.

Adult census: In Figure 1(a), we show the performance of the different
approaches on the Adult census dataset. AdaFair achieves the best (lowest)
statistical parity score, closely followed by AdaBoost SDB. This is also
reflected in the almost identical percentage of positive predictions for the
protected (denoted as Prot. Pos.) and non-protected groups (denoted as Non-
prot. Pos.). In terms of balanced accuracy, we see that AdaFair is only second
best to SMOTEBoost by a 3.5%$\downarrow$ drop (note that SMOTEBoost has the
worst discriminatory behaviour). In comparison to AdaBoost SDB, AdaFair
produces slightly fairer outcomes and 5%$\uparrow$ better performance in terms
of balanced accuracy.

Bank: The results on the Bank dataset are shown in Figure 1(b). AdaFair again
achieves the best fairness score, and in balanced accuracy it outperforms
AdaBoost SDB by a margin of 10%$\uparrow$. SMOTEBoost is 5%$\uparrow$ better
than AdaFair and achieves the best balanced accuracy. AdaBoost achieves the
best performance in terms of accuracy, while all the other methods, including
AdaFair, have similar accuracy. The high accuracy of AdaBoost is because it
predicts most of the instances as negative (the majority class) and very few
instances as positive (the minority) class, which is evident in the last four-
bar plots Prot. Pos, Non-prot Pos, Prot. Neg, and Non-Prot Neg.

Compass: The results on Compass dataset are shown in Figure 1(c). We observe
that AdaFair achieves the best statistical parity score. All the methods have
similar performance in balanced accuracy and accuracy since this dataset does
not suffer from class imbalance. SMOTEBoost and AdaBoost perform very poorly
in terms of statistical parity. AdaFair produces 3.5%$\downarrow$ fairer
outcomes in contrast to AdaBoost SDB, and slightly higher balanced accuracy
(1%$\uparrow$).

KDD census: Figure 1(d) depicts the results on KDD census income dataset. Once
again, we observe that AdaFair achieves the best statistical parity score
compared to the baselines (4%$\downarrow$ lower than AdaBoost SDB). Moreover,
our method AdaFair also achieves higher balanced accuracy than AdaBoost
(8.5%$\uparrow$), AdaBoost SDB (10%$\uparrow$). SMOTEBoost has higher balanced
accuracy (2%$\uparrow$) and high discriminatory outcomes (10%$\uparrow$) in
contrast to AdaFair.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 1: Predictive and fairness, based on Statistical Parity, performance -
higher values are better; for Statistical Parity, lower values are better.

Conclusion: AdaFair performs better than AdaBoost SDB in terms of fairness and
predictive performance. However, SMOTEBoost outperforms AdaFair in balance
accuracy, mainly due to the nature of the fairness notion. Statistical parity
forces AdaFair to shift the decision boundary to achieve parity between the
different groups without considering the true label distribution.

###  5.3 Equal Opportunity: Predictive and fairness performance

In Figure 2, we report on the results for all the approaches w.r.t equal
opportunity. Specifically, we report the predictive performance by accuracy
and balanced accuracy (Bal. Acc.), and also fairness by equal opportunity (Eq.
Op), and TPR and TNR for both protected (Prot.) and non-protected (Non-Prot)
groups.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 2: Predictive and fairness, based on Equal Opportunity, performance -
higher values are better; for Equal Opportunity, lower values are better.

Adult census: In Figure 2(a), we show the results on Adult census dataset. We
observe that our AdaFair achieves the lowest equal opportunity score, which
FAE follows by a minimal margin. We are also marginally better than FAE in
terms of predictive performance, both accuracy and balanced accuracy (around
1%). FAE achieves almost similar performance for both protected and non-
protected groups for both classes. SMOTEBoost achieves the best performance in
terms of balanced accuracy (0.5%$\uparrow$ higher than AdaFair); however, it
is the most unfair model (22%$\uparrow$ higher than AdaFair in terms of equal
opportunity).

Bank: In Figure 2(b), we report on the results of Bank dataset. AdaFair and
FAE have similar equal opportunity scores and balanced accuracy; however, they
behave differently. By examining TPR and TNR for both protected and non-
protected groups, we observe that FAE outperforms AdaFair by 17%$\uparrow$ on
both groups in positive class. However, it deteriorates its performance in the
negative class (17%$\downarrow$ lower TNR in protected group and
10%$\downarrow$ lower TNR in non-protected group). SMOTEBoost produces similar
predictive performance to AdaFair and FAE, but it cannot mitigate unfair
outcomes.

Compass: Figure 2(c) shows the results on Compass dataset. We see that
AdaFair, on this data, produces the fairest results by achieving the minimum
equal opportunity score. By comparing AdaFair to FAE, we observe similar
performance. AdaBoost and SMOTEBoost perform similarly in predictive
performance and discriminatory outcomes since this dataset does not suffer
from class imbalance.

KDD census: The results on KDD census dataset are shown in Figure 2(d). Same
as in the Bank dataset, AdaFair and FAE can mitigate unfair outcomes, but each
method’s outcome w.r.t TPR and TNR of each group are different. Both AdaFair
and FAE outperform SMOTEBoost in terms of balanced accuracy (7%$\downarrow$
and 6.5%$\downarrow$, respectively).

Conclusion: FAE is a method that can compete with AdaFair resulting in similar
equal opportunity and balanced accuracy scores. In case of severe class
imbalance (Bank and KDD census), there is no clear winner: FAE rejects more
negative instances than AdaFair, and AdaFair rejects more positive class
instances. For Adult and Compass, both methods behave similarly.

###  5.4 Disparate Mistreatment: Predictive and fairness performance

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 3: Predictive and fairness, based on Disparate Mistreatment,
performance - higher values are better; for Disparate Mistreatment, lower
values are better.

In Figure 3, we show the results w.r.t disparate mistreatment. For reporting,
we follow the same measures as in equal opportunity, except for fairness, for
which we report on disparate mistreatment ($D.M.$).

Adult census: In Figure 3(a), we show the performance of the different
approaches on the Adult census dataset. SMOTEBoost achieves the highest
balanced accuracy followed by AdaFair ($1$%$\downarrow$); both methods target
class imbalance. The latter, however, also considers fairness. AdaBoost,
Krasanakis et al. and Zafar et al. that do not consider class imbalance have a
7%$\downarrow$, 9%$\downarrow$ and 9%$\downarrow$, respectively, drop in their
balanced accuracy comparing to AdaFair. Regarding disparate mistreatment, as
expected, AdaBoost and SMOTEBoost perform worse as they do not consider
fairness. The best overall disparate mistreatment score is achieved by
Krasanakis et al., followed by our AdaFair ($2\%\uparrow$, recall that lower
values are better). However, a closer look at the actual TPRs and TNRs values
per group show that our method achieves the highest TPRs values for both
protected and non-protected groups compared to the other two fairness aware
approaches. In particular, for the protected (non-protected) group, our TPR is
29%$\uparrow$ (20%$\uparrow$, respectively) higher than the second-best method
of Krasanakis et al. So, it seems that Krasanakis et al. and Zafar et al.
produce low TPRs and high TNRs, i.e., these methods “reject” more positive
class instances to minimize disparate mistreatment (this explains their high
TNRs, low TPRs values). On the contrary, our AdaFair achieves good performance
for both classes (high TPRs, high TNRs) while maintaining good disparate
mistreatment (i.e., low differences in TPRs, TNRs for both protected and non-
protected groups).

Bank: The results are shown in Figure 3(b). All approaches, except for
AdaBoost and SMOTEBoost, achieve low disparate mistreatment. Interestingly,
AdaFair achieves higher balanced accuracy than SMOTEBoost, while it
outperforms the other approaches. A closer look at disparate mistreatment,
namely at TPRs and TNRs, shows significant differences between the approaches.
Namely, w.r.t TPRs, our method outperforms the second-best (Zafar et al.) by
almost 29%$\uparrow$ for each group. Interestingly, AdaBoost and SMOTEBoost
maintain higher TPRs than the methods of Krasanakis et al. and Zafar et al.,
even though they do not consider fairness. The methods of Krasanakis et al.
and Zafar et al. have very similar behaviour, and it seems that both focus on
the majority class (therefore high TNRs, low TPRs). Regarding TNRs, our method
has a small drop of 6%$\downarrow$ and 7%$\downarrow$ drop for the protected
and non-protected groups compared to the second-best approach of Zafar et al.;
this is expected as we optimize for balanced error rather than an overall
error.

Compass: The results are shown in Figure 3(c). Regarding balanced accuracy,
AdaBoost performs best and Zafar et al. worse. However, the differences
between the approaches are not that high. We expect a similar performance of
the different approaches as the dataset is balanced (c.f., Table 1), and
therefore, imbalance treatment has no substantial effect. Regarding fairness,
the method of Krasanakis et al. achieves better performance in terms of
disparate mistreatment (3.5%$\downarrow$) and balanced accuracy
(0.2%$\uparrow$) compared to the second-best AdaFair. Zafar et al. have the
worst disparate mistreatment (almost twice the value of Krasanakis et al.),
recall that its $Bal.Acc.$ was the worse among the approaches. By examining
the TPRs and TNRs of both protected and non-protected groups, we observe that
the performance of Krasanakis et al. is not stable (highest standard deviation
among the methods). Our AdaFair has better TPR values for both groups. Our
TNRs are the lowest among the approaches ($~{}57\%-59\%$) as we optimize for
balanced error, and the negative class represents 54% of the population.

KDD census: We could not use the Zafar et al. approach to this dataset due to
its inability to estimate the optimal parameters. In balanced accuracy,
AdaFair performs 22%$\uparrow$ than Krasanakis et al., 17%$\uparrow$ than
AdaBoost and 5%$\uparrow$ than SMOTEBoost. AdaBoost and Krasanakis et al.
classify almost perfectly the negative class (i.e., TNRs close to 100%), which
comprises 94% of the population. SMOTEBoost has 2%$\downarrow$ to
4%$\downarrow$ drop in TNRs of protected and non-protected groups,
respectively, compared to AdaBoost. Regarding TPR, AdaFair bags the highest
TPR scores for both groups (above 65%) while the method of Krasanakis et al.
results in values below 20%. Both fairness-aware approaches, AdaFair and
Krasanakis et al., minimize discrimination to 2% while AdaBoost and SMOTEBoost
result in 28% and 36% disparate mistreatment, respectively.

Conclusion: AdaFair is capable of achieving high balanced accuracy and low
discrimination by maintaining high TPRs and only slightly worse TNRs for both
groups. On the contrary, the other fairness-aware approaches, namely Zafar et
al. and Krasanakis et al., eliminate discrimination by reducing TPRs, that is,
by rejecting more instances of the positive class to achieve parity among the
protected and non-protected groups. Moreover, Zafar et al. cannot estimate the
optimal parameters for datasets with many attributes (see KDD census failure).

###  5.5 Performance over the boosting rounds

In this section, we analyse the performance of AdaFair over the boosting
rounds. The purpose of this experiment is to show the per round behaviour of
AdaFair w.r.t the objective function, $\theta$ selection and its performance
on the test set. For this analysis, we have selected parameters $c=1$ and
$T=500$ and report on a holdout evaluation (67% training - 33% testing sets)
for each dataset. We report only on disparate mistreatment as it is the most
complex measure comparing to statistical parity and equal opportunity. In
Figure 4, we report on the performance of AdaFair on the test set (only used
for evaluation, not for training) and the validation set, which is used for
selecting the $\theta$ (as mentioned in Section 5.1.3, the validation set is
33% of the training set). We report on the balanced error rate and fairness on
both test and validation set, and also we report on the objective function
(16) w.r.t the validation set.
Adult census: The results are shown in Figure 4(a). We can observe that the
validation set captures the underlying distribution of the test set since they
have almost identical score values. In the early boosting rounds ($T<100$),
AdaFair’s performance fluctuates until it stabilizes. As the number of
boosting rounds increase, the balanced error rate remains the same and the
fairness on the validation set. The values from the objective function
stabilize around $[0.27,0.29]$ for $T>150$. After the training process, we
select the best partial ensemble from weak learners 0 to 488. However, we can
see that the performance on the test set does not change significantly after
$T>100$.
Bank: In Figure 4(b), we report the results on the Bank dataset. Like the
Adult census dataset, high fluctuation can be observed in the early boosting
rounds (e.g., $T<100$). With boosting rounds increasing, the balanced error
rate decreases and stabilises around $0.22$ after $T>400$ (on validation and
test set). The fairness on the validation and test set has some small but
insignificant fluctuations. In this dataset, the gradual decrease of balanced
error rate is the factor that defines $\theta$ since fairness remains low
after $T>100$ (on the validation set).
Compass: In Figure 4(c), we report on the results of Compass dataset. In
contrast to the other datasets, the objective function in Compass fluctuates
for small and high values of $T$. The balanced error rate increases gradually
over the rounds, and fairness fluctuates highly, fluctuating the objective
function. This behaviour is probably caused due to the complexity of the
dataset, e.g., we have seen in Section 5.4 that none of the methods could
achieve more than 70% accuracy or balanced accuracy on this dataset.
KDD census: In Figure 4(d), we report on the results of KDD census dataset.
This dataset shows similar behaviour to the other two imbalanced datasets
(Adult census and Bank). Early boosting rounds ($T<100$) are characterised by
fluctuated values w.r.t predictive performance, fairness and objective
function. After 150 rounds, the performance is stabilised, and we observe a
non-significant decrease in the balanced error rate after 200 rounds.
Conclusions: For the selected datasets, our experiments show that AdaFair
needs at least 100 rounds to produce good results. Although AdaFair does not
tackle class-imbalance in-training, we observe that the balanced error rate
decreases over the rounds. This lies in the fairness-related costs that
indirectly push the model to also learn the minority class. A complete in-
training tackling of fairness and class-imbalance is left for future research.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 4: Analysis over the boosting rounds on the validation and test set;
lower values are better. Fairness measure: Disparate Mistreatment.

###  5.6 Cumulative vs non-cumulative fairness

The notion of cumulative fairness, is crucial for AdaFair’s ability to
mitigate discrimination w.r.t i) statistical parity (Equation (8)), ii) equal
opportunity (Equation (10)), and iii) disparate mistreatment (Equation (12)).
To investigate its impact, we compare AdaFair (with cumulative fairness of
models $1:j$, where $j$ is the current boosting round) with a version that
considers only the fairness of the individual weak learner in round $j$
(refereed to as AdaFair NoCumul), for all the employed fairness notions.
Below, we show the behaviour on disparate mistreatment. The behaviour is
similar across all the employed fairness notions (see Appendix for the exact
outcome on statistical parity and equal opportunity).

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 5:  Disparate Mistreatment: AdaFair vs AdaFair NoCumul

Disparate Mistreatment: The fairness performance w.r.t disparate mistreatment,
for the different datasets is shown in Figure 5. Overall, the AdaFair NoCumul
method results in poor fairness performance with very high $D.M.$ values
compared to AdaFair. In particular, we observe an increase of 78%$\uparrow$
for the Adult census dataset, 59%$\uparrow$ for Bank and 20%$\uparrow$ Compass
datasets, of 80%$\uparrow$ for the KDD census dataset. A closer look at the
individual TPR, TNR scores show that the protected group scores are lower
regarding TPR. In contrast, w.r.t TNR, the scores of the protected group are
higher (we notice opposite behaviour for the Bank dataset). That is, more
protected instances are rejected (low TPR, high TNR). Moreover, the standard
deviation for the non-cumulative version is higher than AdaFair, indicating
AdaFair NoCumul is not stable.

In Figure 6, we compare the per round $\delta FNRs$ and $\delta FPRs$ of the
two approaches. Recall that $\delta FNR$ and $\delta FPR$ define the fairness-
related cost $u$ related to disparate mistreatment fairness that affects the
weighting of the instances for the next round (Equation (13)). Fairness-
related costs of AdaFair NoCumul exhibit a high fluctuation. On the contrary,
the fairness-related costs for our AdaFair are smoother and converge after a
sufficient number of rounds to a specific range $[-0.05,0.05]$. That means
that our method mitigates discrimination over the early rounds. These results
further confirm that the cumulative definition of fairness is superior to a
non-cumulative approach.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 6: Disparate Mistreatment, fairness-related costs per boosting round:
AdaFair vs AdaFair NoCumul

Conclusion: By considering the cumulative fairness of the ensemble, AdaFair
can produce fairness-related costs which are more stable than its non-
cumulative version. These costs allow the model to find a hypothesis that
mitigates discriminatory outcomes significantly better than non-cumulative
costs.

###  5.7 The effect of balanced error

AdaFair can achieve fairness even when it does not optimize for a balanced
error rate because of the cumulative fairness notion that alters the data
distribution during training in the direction of fairness. In this section, we
show that varying the parameter $c$, which alternates the objective goal
(Equation (16)), does not deteriorate the ability of AdaFair to mitigate
discriminatory outcomes. We show the impact of parameter $c$ for all the
employed fairness notions in Figure 7. In these figures, we plot accuracy and
fairness related measures for different values of $c\in[0,1]$. For $c=1$, the
balanced error is optimized while for $c=0$, the error rate is optimized.
Values in-between (we use a step of 0.2 for $c$) correspond to different
balanced error and error rate combinations. We report on the average of 10
random splits for each value of $c$. Same as previously, we show only the
results w.r.t disparate mistreatment and urge the interested readers to see
the Appendix for the results on other fairness notions.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 7: Disparate Mistreatment: impact of parameter $c$

Disparate Mistreatment: As we can see in Figure 7, for all imbalanced datasets
(Adult census, Bank, KDD census), the balanced accuracy increases with $c$.
For $c=0$ (only error rate is considered), the TPRs for both groups are very
low. The TPRs increase with $c$, reaching their best values at $c=1$, i.e.,
when balanced accuracy is considered. TNRs decrease with $c$, though their
decrease is lower than the increase of TPRs. This again supports our previous
findings that AdaFair achieves parity between the two groups for both TPRs and
TNRs while achieving high TPRs, on the contrary to Zafar et al. and Krasanakis
et al. (c.f., Section 5). For Compass, accuracy and balanced accuracy are very
close, and therefore no significant differences are observed by varying $c$.
More precisely, a comparison of the results for $c=0$ (error rate) and $c=1$
(balanced error rate) shows i) for the Adult census dataset, 12%$\uparrow$ and
11%$\uparrow$ increase in TPRs (for $s$, $\bar{s}$ groups respectively) and
4%$\downarrow$ and 5%$\downarrow$ reduction in TNRs (for $s$, $\bar{s}$ groups
respectively); ii) for the Bank dataset, 16%$\uparrow$ increase in TPRs (for
both groups) and only 1%$\downarrow$ and 2%$\downarrow$ decrease in TNRs (for
$s$ and $\bar{s}$, respectively) and iii) for the KDD census dataset,
27%$\uparrow$ growth in TPR (for both groups) and only 4%$\downarrow$
reduction in TNRs (for both groups).

Conclusion: The parameter $c$ is essential in the presence of class imbalance
since it can help the user to examine the trade-off between TPRs and TNRs
without losing AdaFair’s ability to mitigate discriminatory outcomes for any
given fairness measure. Moreover, in balanced datasets, any value of parameter
$c$ ($\in[0,1]$) is indifferent since balanced accuracy and accuracy are
similar.

##  6 Conclusions and Future Work

We propose AdaFair, a fairness-aware boosting approach that adapts AdaBoost to
fairness by changing the data distribution in each round based on both model
error and cumulative fairness performance. Moreover, at post-training, AdaFair
selects the best sequence of weak learners that optimizes for balanced error
performance. The notion of cumulative fairness evaluates in each round the
fairness-related behaviour of the current partial ensemble and adjusts the
fairness-related costs of the discriminated group accordingly. We introduce
the cumulative counterparts of three popular parity-based fairness notions,
namely: statistical parity, equal opportunity, and disparate mistreatment and
we provide an in-depth error analysis of AdaFair for each case. Our
experiments show that AdaFair can mitigate discriminatory outcomes while
maintaining good predictive performance across both classes, even for datasets
with severe class-imbalance.

A possible extension, already discussed in Section 4.2.2, is the direct in-
training mitigation of class imbalance. Another interesting direction is the
online selection of the optimal number of boosting rounds $\theta$. Finally,
we plan to investigate further the notions of cumulative fairness building
upon ideas on combined discrimination from the legal domain [41] .

## Acknowledgements

The work is supported by the Volkswagen Foundation project BIAS (”Bias and
Discrimination in Big Data and Algorithmic Processing. Philosophical
Assessments, Legal Dimensions, and Technical Solutions”) within the initiative
”AI and the Society of the Future”.

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## Appendix

### Cumulative vs non-cumulative fairness

Statistical Parity: In Figure 8, we show the comparison of AdaFair versus
AdaFair NoCumul w.r.t statistical parity for each dataset. As we see, AdaFair
NoCumul produces higher discriminatory outcomes than AdaFair on all datasets.
For the Adult census dataset, we observe a 31%$\uparrow$ increase,
12%$\uparrow$ increase for the Bank dataset, 15%$\uparrow$ for the Compass and
15%$\uparrow$ for the KDD census dataset. The cumulative notion of fairness
allows AdaFair to effectively mitigate the discriminatory outcomes in contrast
to the non-cumulative version.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 8:  Statistical parity: AdaFair vs AdaFair NoCumul

In Figure 9, we compare the per round $\delta SP$ of AdaFair NoCumul and
AdaFair. $\delta SP$ refers to the fairness-related cost ($u$) that is
assigned to instances based on the discriminatory behaviour of the model
(Equation (9)). We observe that AdaFair NoCumul produces fairness-related
costs, which highly fluctuate, in contrast to AdaFair, in all the datasets.
The non-cumulative version cannot stabilize the fairness-related costs since
it depends on the behaviour of individual weak learns rather than the
cumulative behaviour of the model.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 9: Statistical parity, fairness-related costs per boosting round:
AdaFair vs AdaFair NoCumul

Equal Opportunity: In Figure 10, we show the comparison of AdaFair versus
AdaFair NoCumul w.r.t equal opportunity for each dataset. Same as in the
statistical parity case, AdaFair NoCumul produces more discriminatory outcomes
in contrast to AdaFair. For the Adult census dataset, there is a 15%$\uparrow$
increase, 2%$\uparrow$ increase for the Bank dataset, 12%$\uparrow$ increase
for the Compass, and 8%$\uparrow$ increase for the KDD census dataset.

Similar behaviour to statistical parity is also observed in Figure 11, where
we report $\delta FNR$ values for the cumulative and non-cumulative
approaches; $\delta FNR$ values are employed as fairness-related costs and are
derived from Equation (11). The non-cumulative version is unstable and
produces highly fluctuating fairness-related costs in contrast to AdaFair in
all datasets.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 10:  Equal opportunity: AdaFair vs AdaFair NoCumul

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 11: Equal opportunity, fairness-related costs per boosting round:
AdaFair vs AdaFair NoCumul

### The effect of balanced error

We show the impact of parameter $c$ for all the employed fairness notions in
Figures 12 and 13.

Statistical Parity: In Figure 12, we show the impact of parameter $c$ in case
of statistical parity. As we observe, all the imbalanced datasets show the
worst performance in terms of balanced accuracy when $c=0$; however,
statistical parity is close to 0. As the parameter $c$ increases, the balanced
accuracy increases and the statistical parity remains close to 0. However, in
the case of statistical parity, we observe that the balanced accuracy is not
affected significantly in contrast to the other two fairness notions. Such
behaviour is caused due to the fairness’ notion, which forces parity between
protected and non-protected groups on the predicted outcomes; thus,
statistical parity can force AdaFair to predict more instances in the positive
class indirectly.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 12: Statistical Parity: impact of parameter $c$

Equal Opportunity: In Figure 13, we show the impact of $c$ when AdaFair tunes
for equal opportunity. Similar to disparate mistreatment, AdaFair can maintain
its low discrimination values w.r.t equal opportunity and at the same time
increase the balanced accuracy as the parameter $c$ increases. E.g., AdaFair’s
balanced accuracy increases 8% for $c=0$ to $c=1$ and at the same time equal
opportunity is close to 0. This behaviour is similar for all the employed
imbalanced datasets. For the Compass dataset, the parameter $c$ does not
affect the performance significantly since the dataset is class balanced.

(a) Adult census

(b) Bank

(c) Compass

(d) KDD census

Figure 13: Equal Opportunity: impact of parameter $c$