Fair Learning by Model Averaging

Abstract

The central question of this paper is how to simply enhance a set of given supervised learning algorithms under some fairness requirements, to ensure that any sensitive variable does not “unfairly” influence the outcome. To achieve this goal, we work with several notions of fairness (Demographic Parity, Equalized Odds, Lack of Disparate Mistreatment), possibly generalised to more general concepts of conditional fairness. We linearly combine an ensemble of binary and/or continuous basis classifiers or regressors to build an “approximately optimal” solution in terms of fairness and accuracy for any given notion of fairness. The trade-off between fairness and predictive power is managed by considering penalised criteria. We rely on post-processing procedures without any transformation of the data nor of the basis training algorithms. Some empirical experiments, by simulation and on real databases, are provided to illustrate our approach.

Keywords

Ethics fairness classifiers bagging penalised criteria

1. Introduction

Machine learning is increasingly used to take decision that can severely affect people’s life in education, living, working, lending or criminal risk assessment, for instance. This phenomena has been accompanied by an increase in concern about disparate treatment caused by potential model failures and patent biases in training data. The potential for unfairness¹ in algorithmic decisions being growing, an important recent stream of the literature around some notions of fairness and fair algorithms is rapidly developing. It has been inspired by the concept of discrimination in social sciences and law, extending previous works done for instance by Cleary (1966) and Einhorn and Bass (1971), but under the point of view of machine learning. To be short, to ensure that learned classifiers (or regressors, more generally) are non-discriminatory - also called “fair” - with respect to some sensitive feature has become a crucial question.

A significant effort in the machine learning community (and beyond) has focussed on the development of mathematical definitions of fairness (Berk et al., 2018; Hardt et al., 2016; Madras et al., 2018, among others) and of algorithmic methods to assess and mitigate biases in relation to these definitions: Agarwal et al. (2018), Kusner et al. (2017), Zafar et al. (2017), Lundberg and Lee (2017), among others. The interest in improving fairness for classification or regression purpose is to be able to make “responsible” predictions. A model should be considered as convenient under this ethical point of view if its predictions are reliable and aligned with the system objectives, which most often includes accuracy (predictions should mostly indicate ground truth) and fairness (predictions should be unbiased with respect to different subgroups).

For a given notion of fairness, a challenge is then to propose some modelling procedures in order to get more fair predictors without sacrifying too much accuracy (predictive power). Numerous ways of obtaining fairness - at least approximately - are reviewed in Section 2. In this paper, we will adopt a post-processing point-of-view: for some given notion of fairness, once one or several predictor(s) are calibrated, their performance(s) could be adjusted and/or combined to obtain more fair predictions. We will start from some machine learning “basis” algorithms that will be applied to the same dataset. This is not demanding because several simple and/or standard methods for classification/regression purpose are most often available in practise. In order to provide fair predictions, we linearly combine the latter set of predictors, in the spirit of random forests (Breiman, 2001). Usual ensemble learning algorithms invoke the same type of basis classifiers to produce repeated multiple classifications of the same records (Kotsiantis & Pintelas, 2004). In our case, we will potentially use very different basis classifiers/regressors, and we will require only a few ones instead of hundreds (at least), as for random forests. Our challenge will be to calibrate the respective weights of some basis predictors in an “optimal” way, depending on the notion of fairness we have selected. This will be done by optimising some penalised criteria, after having empirically estimated some convenient measures of fairness and accuracy. Our final predictors will be linear combinations of the basis predictors, whose weights depend on a trade-off between accuracy and fairness (through the choice of an hyperparameter). In some classical situations, we are even able to exhibit closed-form solutions.

The rest of the paper is organised as follows. Section 2 provides an analysis of the literature about fairness definitions and the ways of obtaining more fair predictors. In Section 3, we formally define all the fairness measures we investigate, i.e., demographic parity, equalised odds and lack of disparate mistreatment. Moreover, we introduce conditional versions of such measures, given an arbitrarily chosen set of explanatory variables. In Section 4, we introduce our new post-processing procedure, i.e. building several predictors by model averaging. Where possible, optimal solutions are provided, depending on the selected notion of fairness. The trade-off between fairness and accuracy is managed through penalisation. Some numerical experiments on real datasets and by simulation illustrate the relevance of our approach in Section 5 and in Section A in the appendix.

2. Analysis of the Literature

The related machine learning literature has tended to focus on domains such as recidivism prediction, automated hiring and face recognition, where fairness can be understood, at least partially, in terms of well-defined quantitative metrics. It has been argued that some algorithms trained with biased data may result in algorithmic discrimination, as in illustrated in Barocas et al. (2023). For example, some statistical methods used in the US judicial system are questioned, pointing to a bias against African-Americans. Indeed, African-American defendants were more likely to be wrongly classified as having a higher risk of recidivism. Thus, a fairly extensive literature has proposed different criteria to identify and correct such biases. All these works have been enriched by research on discrimination involving gender, race, colour, religion, disability, family status, children health, facial recognition, etc. For instance, Buolamini and Gebru (2018) analyze pictures to evaluate how biases can be present in automated facial analysis algorithms and datasets with respect to phenotypic subgroups. Chouldechova et al. (2018) investigate the decision of children’s placement in case of abuse based on administrative information and how such pieces of information can impact decisions. Other examples can be found in O’Neil (2016), Noble (2018) and Lambrecht and Tucker (2019), among others.

The simplest and most intuitive notion of fairness is statistical parity, also called group fairness or demographic parity, which equalizes outcomes across protected and non protected groups. Demographic parity of a predictor requires that its predictions are independent of a protected attribute (also called “sensitive attribute”). Under this approach, the (unconditional) distributions of the predicted class/outcome should be the same for both protected group categories. From an intuitive viewpoint of justice, a strict application of demographic parity can create highly undesirable decisions. For instance, if the protected attribute is gender, one might incarcerate women who pose no public safety risk so that the same proportions of men and women are released on probation. This problem has been illustrated in Dwork et al. (2012).

In order to avoid the limitation of the previous concept of fairness, it seems necessary to take into account the “true” output, beside its predictor and the protected attribute. This induced the idea of equalised odds with respect to a protected attribute. This occurs when the given predictor and the protected attribute are independent conditionally on the output (Hardt et al., 2016). This notion of fairness is also called conditional procedure accuracy equality (Berk et al., 2018). Thus, a binary predictor satisfies equalised odds if two individuals in different groups are treated comparably by this predictor when they exhibit comparable future behaviour.

The closely related concept of equal opportunity boils down to equalised odds when focusing on a particular value of the discrete outcomes only. This notion is used in particular when one specifies a decision concerning a beneficial outcome, like “not defaulting to a loan” or “admission to a college”, for instance. This notion is called balance for the positive (or negative) class in Kleinberg et al. (2017). Beside, another unfairness metric of interest, called disparate mistreatment, has been introduced by Zafar et al. (2017). These authors call a decision making process to be suffering from disparate mistreatment with respect to a given sensitive attribute (e.g., race) if the mis-classification rates differ for groups of people having different values of that sensitive attribute (e.g., black or white).

In our work, we will develop our methods by working with the concepts of demographic parity, equalised odds and the lack of disparate mistreatment, which are among the most popular and widely accepted (Verma & Rubin, 2018, Table 1).

Table 1.
Summary of the “Adult” and “COMPAS” Datasets.

Name Nb of Observations Sensitive Feature Target Variable

Adult 45222 gender income > 50k USD

COMPAS 6172 race recidivism

Name	Nb of Observations	Sensitive Feature	Target Variable
Adult	45222	gender	income > 50k USD
COMPAS	6172	race	recidivism

The idea of improving the fairness of classifiers/regressors is not new and several strategies have emerged in the literature. Indeed, in order to address the ethical problem posed by a lack of fairness, several fairness-aware machine learning algorithms have been investigated. As in Friedler et al. (2019), they can be categorised as (i) pre-processing techniques, designed to modify training data so that the outcomes of any machine learning algorithm applied to such data may be considered as fair; (ii) algorithm modification techniques, that modify an existing algorithm or create a new one that will be fair for any traning dataset; (iii) post-processing techniques, that rework the outputs of some model so that it becomes more fair.

The motivation behind pre-processing algorithms is the idea that training datasets may be a source of discrimination that a machine learning algorithm might learn. This phenomenon could appear because the training data itself captures historical discrimination, or because there are more subtle patterns in the data (under-representation of a minority group, e.g.). In order to remove the assumed bias introduced by a dataset, the first natural idea would be to modify the labels of some attributes. For instance, this means that the proportions of positive labels are equal in the protected and unprotected groups. A classifier is then trained with these new labels assuming that equal opportunity of positive labelling will become widespread in test sets. Calders et al. (2009) and Kamiran and Calders (2012) propose diverse approaches to attain this objective: removal of some raw data (based on correlations analysis), change of their labels, re-weighting of observations. Calmon et al. (2017) have chosen to work by randomisation, following the works of Pedreschi et al. (2009) and Zemel et al. (2013). Another idea was proposed by Dwork et al. (2012) for statistical parity. It consists in mapping training datasets to intermediate representations, by optimising a classification criterion while constraining nearby individuals to be mapped similarly. A major difficulty with pre-processing algorithms comes from interaction effects (e.g., between race and earnings). Some a priori non sensitive variables may contain information leading to unfairness. In theory, all interactions effects between senstive features and the others would need to be anticipated and removed. Obviously, even if it were possible, this cleaning procedure would induce a significant loss of information. Rebalancing the information set is another possibility (Berk et al., 2018; Zemel et al., 2013). Lu et al. (2017) use actionable plans to transform predictions of the inputs to desired outputs at minimal cost.

Beside, algorithm modifications are present in the form of additional constraints during the calibration stage, considering that removing sensitive features is insufficient. This is most often made through regularisation or constrainted optimisation. For instance, Kamishima et al. (2012) introduce a fairness focussed regularisation term and apply it to a logistic regression classifier. Considering disparate impact fairness, Feldman et al. (2015) impose to their binary classifier a constraint that corresponds to a balanced error rate. Calders and Verwer (2010) build separate models for each value of a sensitive attribute and apply the appropriate model depending on the value of this attribute. Zemel et al. (2013) combine pre-processing and algorithm modification. Their approach is to learn a modified representation of data that is most effective for classification purpose, while still being free of signals pertaining to the sensitive attribute. Observing that standard fairness constraints are non convex and cannot be easily satisfied, Zafar et al. (2017) introduce a convex relaxation during an optimisation stage. Following the same ideas, Donini et al. (2018) directly enforce fairness during the training step of a classifier, optimising a fairness constraint related to the notion of equalised odds. See Kamishima et al. (2012), Agarwal et al. (2018), among others, and a review by Friedler et al. (2019). In some cases, optimal fair classifiers can be explicitly obtained on a theoretical basis: see Chzhen et al. (2020) and del Barrio et al. (2020).

A third approach to obtain fairness in Machine Learning is by modifying the results of a previously trained predictor to achieve the desired results on different sub-groups. These ideas are categorised as post-processing techniques, even if they often may be seen as algorithm modifications too. For example, Kamiran et al. (2010) propose an approach in which the non-discriminatory constraint is deeply pushed into a decision tree learner by changing its splitting criterion and its pruning strategy. They use a novel leaf re-labelling approach after training in order to satisfy fairness constraints. A similar approach is proposed in Zliobaite (2015) who splits the initial dataset, trains a logistic regression, outputs class probability scores for the test set and varies the classification threshold from 0 to 1, which changes the acceptance rate. Hardt et al. (2016) propose to enforce equalised odds binary choice situations based on univariate scores by conveniently choosing thresholds. They introduce a predictive score with the enforcement that higher values of this score correspond to greater likelihoods of the outcome of interest. A binary classifier is obtained by thresholding the score. A score satisfies equalised odds if it is independent of the sensitive feature given the outcome. The idea that the prediction error restricted to any protected group remains below some pre-determined level has also be investigated in a regression setting under statistical parity by Agarwal et al. (2019). See also a related work by Chzhen et al. (2020), in which the authors measure unfairness by the Total Variation distance in place of Kolmogorov-Smirnov distance. Woodworth et al. (2017) explored the use of post-processing as a way to ensure fairness with respect to error profiles by directly incorporating non-discrimination into the learning process. A similar approach is developed in Kearns et al. (2018) who work with subgroups in place of individuals. A random reassignment of the class labels previously assigned by the algorithm has been proposed in Feldman et al. (2015). Agarwal et al. (2018) used a weighted classification implementation of logistic regression and gradient-boosted trees in order to find the lowest-error distribution over a class of classifiers.

Our approach by model averaging is different from the previous methods due to several points. First, it cannot be associated with a pre-processing technique : we do not modify the input data to make fair the outcomes of the classifiers applied to the data. Indeed, the motivation behind preprocessing algorithms is the idea that training data is the cause of discrimination in most machine learning algorithms. And, by modifying data, one can protect most learning algorithm from the risk of discrimination. However, to avoid this pitfall, one must be sure that all correlations (or other more subtle relationships between the features in the data) have been removed. This requires having “anticipated” all the hidden relationships between these variables, a very difficult and questionable task. Second, our technique by “averaging” is not related to algorithm modification because it does not rework the calibration stage of our basis classifiers/regressors. Indeed, reworking the basis predictors under fairness constraints in terms of calibration may be a source of worry in practical terms (numerical burden, recoding, etc), and their fairness woud be enforced only approximately most of the time.

Actually, the closest paradigm to our method by model averaging is post-processing. Indeed, we propose to linearly combine several “basis predictions” that are generally unfair to obtain more fair predictions, and without sacrificing too much predictive power. Our technique does not require to build new estimators but simply to combine the existing ones conveniently, a clear advantage in practise. Moreover, by taking such basis predictors as given and built in a preliminary stage that we do not reconsider, we can combine potentially very-complex basis predictors (neural networks, in particular) without opening the “black-box”. Note that averaging a group of classifiers/regressors generally performs more accurately than using a single one, by combining the strengths of the individual predictors - which can be very different - while at the same time their weaknesses are mitigated. The latter idea is applied in machine learning under the terminologies “bagging” or “ensemble methods”. This is recognized as a powerful way of decreasing prediction errors (Hastie et al., 2009, Section 8).

There are no direct connexions between our way of working and the existing post-processing techniques that have already been proposed in the literature. Nonetheless, independently and in parallel to this work, some authors have recently applied similar ideas, but restricting themselves to a particular family of machine learning models or/and in a different spirit. Grgić-Hlaa et al. (2017) were the first to note that ensemble method can improve model fairness. They propose using an ensemble method that randomly selects a classifier from a pool. Bhaskaruni et al. (2019) have reworked the AdaBoost weighting scheme to improve fairness, using the k-NN technique of unfairness detection proposed by Luong et al. (2011). See Iosifidis et al. (2019) too, that apply Adaboost in addition to post-processing. Kenfack et al. (2021) assign weights of classifiers proportionally to their performances in term of fairness and accuracy, respectively evaluated in terms of an arbitrarily chosen fairness metric and classification errors. Nonetheless, their reweighting formula may be seen as rather arbitrary. To improve fairness, Zhang et al. (2021) proposed the so-called Fair and Adaptive Random Forests (FARF), an adaptation of the classical random forest algorithm. Still in the case of random forests, Fantin (2020) and Hu et al. (2019) assume the existence of a third party which holds private demographic information, audits the (numerous) trees generated from a data centre that stores non-private data only, and selects those that are acceptable in terms of fairness. The final predictor is finally a weighted average of the retained individual predictors. The trade-off between fairness and accuracy was considered by Guardieiro et al. (2023) as a multi-objective optimisation problem. To exhibit Pareto-optimal models, they propose an iterative calibration strategy for finding weights and model parameters. On our side, the model parameters are fixed and only the weights have to be found. See other references in Guardieiro et al. (2023). Note that many so-called “fair” decision trees - including random forests - techniques have also been proposed in the literature. Most of them propose to modify the information gain metric when selecting attributes to split leaf nodes in a fair way. With our terminilogy, they belong to the family of algorithm modifications: see Raff et al. (2018), Nanfack et al. (2021), Zhang et al. (2021), Abebe et al. (2022), among others.

Finally, even if most of the reviewed literature comes from computer science, there are also many recent contributions in economics and econometrics on this topic. Some of these papers incorporate fairness through some constraints and/or penalties as in our work. Others state a welfare function as a loss, by building group-specific weights for instance. Such welfare functions yield interesting alternative approaches to penalisation method based on misclassification errors and MSE. See Rambachan et al. (2020), Ghysels et al. (2020), Kitagawa et al. (2021) and references therein.

3. Preliminaries: Definitions of Fairness

This preliminary section formally recalls all definitions and concepts of fairness that will be used afterwards. All of them have already been defined in the literature of the domain.

Let $X$ be a vector of individual characteristics (exogenous variables) in a finite dimensional space $X$ . We want to explain/predict a univariate random variable $Y$ , that can be discrete or continuous. Beside, a variable $S$ is considered as a sensitive feature: gender, ethnical background, etc. In line with the literature, the latter sensitive feature is discrete. To simplify, it will take only two values (zero or one). The sensitive feature $S$ will be a component of $X$ , but $X$ may contain other features, that may be more or less indicative of $S$ . For example, if the classification task is to predict whether or not someone will go to default on a loan, each training record might correspond to a person, and $X$ represents his/her socio-demographic indicators, income level, past payment history, and loan amount; $S$ could represent “race”. Moreover, $Y$ indicates whether or not such an individual will be defaulted on that loan. Note that, even if $X$ might not contain the information “race” as one of the features, it could contain features that are often correlated with race (zipcodes, for instance). Our aim is to build a predictor of $Y$ given $X$ that is “fair” w.r.t $S$ , without changing the values of the input variables.

Strictly speaking, a (univariate) predictor is a mapping $g : X \to R$ that seeks to predict $Y \in R$ from some individual characteristics $X$ , possibly including $S$ . Denote by $\hat{Y}$ such predictions, namely $\hat{Y} := g (X)$ for any $X$ . Hereafter, all probabilities and expectations will be related to the random vector $(X, Y)$ and the maps $g$ are assumed to be deterministic (and known). In other words, the fact that the latter predictors are generally obtained by other (first stage) statistical procedures will not be considered as an additional source of randomness in our framework.

As discussed in the introduction, many definitions of fairness have been proposed in the literature: see Williamson and Menon (2019), and the references therein. Now, we mathematically define the concepts that will be of interest in this paper.

(i)
Demographic parity (also called statistical parity or DP) is satisfied when $\hat{Y}$ and $S$ are statistically independent. If we assume that the random variable $\hat{Y}$ is discrete and potentially takes $p$ values in ${0, 1, \dots, p - 1}$ , then, demographic parity is equivalent to the property
$P (\hat{Y} = j | S = k) = P (\hat{Y} = j), j \in {0, \dots, p - 1}, k \in {0, 1} .$
(3.1)
When $p = 2$ , this is equivalent to $P (\hat{Y} = 1 | S = 0) = P (\hat{Y} = 1 | S = 1)$ , and equation (3.1) reduces to $E [\hat{Y} | S = 0] = E [\hat{Y} | S = 1]$ = $E [\hat{Y}]$ because $\hat{Y} \in {0, 1}$ . To illustrate, if $S$ represents gender and Y the recidivism variable, one wants that the probability of recidivism of a person is the same given that this person is a male or is a female.²

In the case of a continuous predicted variable $\hat{Y}$ , demographic parity may be rewritten in terms of equality between distributions, i.e.
$P (\hat{Y} \leq y | S = 0) = P (\hat{Y} \leq y | S = 1) = P (\hat{Y} \leq y),$
(3.2)
for every real number $y$ . In this case, this implies (but is not equivalent to)
$E (\hat{Y} | S = 0) = E (\hat{Y} | S = 1) = E (\hat{Y}) .$
(3.3)
The latter necessary moment condition is less demanding and more convenient for inference/calibration purpose than (3.2). Focusing on (3.3) instead of (3.2) is particularly relevant in the case of regression-type fair models, as in Centorrino et al. (2022). Indeed, in such cases, models themselves are defined in terms of conditional expectations, which makes it easier to take fairness constraints of the type (3.3) into account.

While simple and intuitive, demographic parity has serious conceptual limitations as a fairness notion, many of which are pointed out in Dwork et al. (2012).
(ii)
Equalized odds (EO) is satisfied if $\hat{Y}$ and $S$ are statistically independent, given $Y$ . Assume that the random variables $Y$ and $\hat{Y}$ are discrete and potentially takes $p$ values in ${0, 1, \dots, p - 1}$ . Then, equalised odds reduces to
$P (\hat{Y} = j | S = k, Y = l) = P (\hat{Y} = j | Y = l), j, l \in {0, \dots, p - 1}, k \in {0, 1} .$
When $p = 2$ (binary explained variables), this is equivalent to
$E [\hat{Y} | S = 0, Y = l] = E [\hat{Y} | S = 1, Y = l] = E [\hat{Y} | Y = l], l \in {0, 1} .$
Assuming $S$ represents gender and Y recidivism, EO means that the mean predicted probability of recidivism of a person is the same knowing that this person is a male or a female and he/she reoffends. Obviously, EO is well-suited for scenarios where the true outcome $Y$ is observable, i.e. where the so-called “ground truth” exists. In the case of continuous explained variables $Y$ and $\hat{Y}$ , equalised odds may be rewritten as
$P (\hat{Y} \leq y_{1} | S = 0, Y = y_{2}) = P (\hat{Y} \leq y_{1} | S = 1, Y = y_{2}) = P (\hat{Y} \leq y_{1} | Y = y_{2}),$
(3.4)
for every real numbers $y_{1}$ and $y_{2}$ . This implies
$E [\hat{Y} | S = 0, Y = y] = E [\hat{Y} | S = 1, Y = y] = E [\hat{Y} | Y = y],$
(3.5)
for every $y \in R$ . Note that (3.4) is equivalent to (3.5) only when $\hat{Y}$ is binary.

When the random variables $Y$ and $\hat{Y}$ are discrete, binary $0$ - $1$ and if we favour the event ( $Y = 1$ ) - the “advantaged” outcome, for instance “receiving a promotion” or “not defaulting on a loan”-, then equal opportunity (Hardt et al., 2016) is satisfied when we have
$E [\hat{Y} | S = 0, Y = 1] = E [\hat{Y} | S = 1, Y = 1] = E [\hat{Y} | Y = 1] .$
Thus, EO implies equal opportunity but not the opposite. Remark 3.1
All our definition and further developments around equalised odds can be easily adapted to the related concept of “sufficiency” (Barocas et al., 2023), for which the roles of $Y$ and $\hat{Y}$ are formally switched. In other words, when $\hat{Y}$ and $S$ were independent given $Y$ for equalised odds, the triplet of random variables $(\hat{Y}, S, Y)$ is said “sufficient” when $Y$ and $S$ are independent given $\hat{Y}$ , yielding an alternative concept of fairness.

(iii)
In the case of discrete outcomes, the lack of disparate mistreatment (LDM) (Zafar et al., 2017) is defined as
$P (\hat{Y} \neq Y | S = 0) = P (\hat{Y} \neq Y | S = 1) .$
Contrary to the latter concepts, we now focus on prediction mistakes, that have to be “similar” whatever the value of the sensitive feature. Thus, in order to avoid disparate mistreatment, an usual approach for model calibration is based on the minimisation of a convex loss under the constraint $| P (\hat{Y} \neq Y | S = 0) - P (\hat{Y} \neq Y | S = 1) | \leq ε$ for some $ε > 0$ , and the smaller $ε$ is, the more fair the decision boundary would be. With binary outcomes, the latter definition is equivalent to
$E [| Y - \hat{Y} |^{α} | S = 0] = E [| Y - \hat{Y} |^{α} | S = 1],$
(3.6)
for some constant $α > 0$ . Dealing with conditional laws for the explained variable, lack of disparate mistreatment would be a consequence of the relationship
$P (Y - \hat{Y} \leq y | S = 0) = P (Y - \hat{Y} \leq y | S = 1), y \in R .$
(3.7)
Then, it makes sense to extend the concept of lack of disparate mistreatment for continuous outcomes through the relationships (3.6) or even (3.7). Note that the latter definition is stronger than the former: (3.7) means an equality in law, when (3.6) is related to an equality in expectations. Such concepts will be called strong and weak lack of disparate mistreatment respectively.

The discrepancy between $Y$ and $\hat{Y}$ given $(S = 0)$ and/or $(S = 1)$ is at the heart of LDM. To inspect more closely-related but more peculiar features, Zafar et al. (2017) proposed to check whether the following relationships apply, in the case of discrete outcomes:
$P (\hat{Y} \neq Y | S = 0, Y = 1) \neq P (\hat{Y} \neq Y | S = 1, Y = 1)$ (fairness bias by false negative signals);

$P (\hat{Y} \neq Y | S = 0, Y = 0) \neq P (\hat{Y} \neq Y | S = 1, Y = 0)$ (fairness bias by false positive signals);

$P (\hat{Y} \neq Y | S = 0, \hat{Y} = 1) \neq P (\hat{Y} \neq Y | S = 1, \hat{Y} = 1)$ (fairness bias by false discovery signals);

$P (\hat{Y} \neq Y | S = 0, \hat{Y} = 0) \neq P (\hat{Y} \neq Y | S = 1, \hat{Y} = 0)$ (fairness bias by false omission signals).
Other references are Woodworth et al. (2017) and Darlington (1971).
(iv)
The criteria above encourage us to consider fairness based on conditional probabilities/expectations/laws. Now, we define a general notion of conditional fairness, that extends any previously defined concept of fairness. Assume that $X$ includes the sensitive feature $S$ plus some other variables, among which some of them are of particular interest. To fix the ideas, $X = (Z_{1}, Z_{2}, S)$ and $S$ does not belong to $Z := (Z_{1}, Z_{2}) \in Z := Z_{1} \times Z_{2}$ , for some subsets $Z_{1} \subset R^{q}$ and $Z_{2} \subset R^{p - q}$ . We focus on the sub-vector of covariates $Z_{1}$ that will be the set of conditioning variables “of interest”. Now, a predictor will be considered as fair when adding the information about the sensitive feature $S$ does not change the prediction of $Y$ , given any value of the sub-vector $Z_{1}$ . Generally speaking, a predictor will be said conditionally fair given $Z_{1}$ if the sensitive feature does not bring any additional information to the prediction of $Y$ by $\hat{Y} (X)$ , for any value of $Z_{1}$ and for a given concept of fairness. Generally speaking, conditional fairness removes illegal discrimination, allowing the disparity in decisions to be present as long as they are explainable (Corbett-Davies et al., 2017).
In the case of demographic parity and discrete outcomes, this means
$P (\hat{Y} = j | S = 0, Z_{1} = z_{1}) = P (\hat{Y} = j | S = 1, Z_{1} = z_{1}), j \in {0, \dots, p - 1}, z_{1} \in Z_{1} .$
In particular, the latter relationships imply
$E [\hat{Y} | S = 0, Z_{1} = z_{1}] = E [\hat{Y} | S = 1, Z_{1} = z_{1}] = E [\hat{Y} | Z_{1}], z_{1} \in Z_{1} .$
For example, a judicial process can be considered as fair relatively to ethnical background given earnings, if the probability of being convicted is the same for a black person and a white person with the same income level. In terms of terminology, we call such stronger properties with the adjective “conditional”. Here, we would say that the judicial process is fair in terms of conditional demographic parity given earnings. Simply, DP has to be fulfilled given every conditioning events $(Z_{1} = z_{1})$ for any $z_{1} \in Z_{1}$ . Endowed with this additional information about the “not sensitive” features $Z$ (including the interesting features $Z_{1}$ ), it is possible to revisit all the notions of fairness that were introduced in (i)-(iii). In particular, a binary classifier does not suffer from conditional disparate mistreatment given $Z_{1}$ if, for every $z_{1} \in Z_{1}$ ,
$P (\hat{Y} \neq Y | Z_{1} = z_{1}, S = 0) = P (\hat{Y} \neq Y | Z_{1} = z_{1}, S = 1) = P (\hat{Y} \neq Y | Z_{1} = z_{1}) .$
This means that the probability of misclassification does not depend on the sensitive feature $S$ , for any given value of $Z_{1}$ .

All the concepts of conditional fairness impose stronger constraints than the previously defined criteria.³ Even if the original idea of conditional fairness already appeared in Kamiran et al. (2013), it formal definition emerged only recently. Notably, “conditional statistical parity” has been defined in Corbett-Davies et al. (2017), Ritov et al. (2017), Xu et al. (2020), Hurlin et al. (2024). In this paper, we extend this concept beyond DP and towards EO and LDM.
4. Fair Classifiers/Regressors by Model Averaging

We present our new class of fair predictors by “model averaging” (or “bagging”), for the different notions of fairness we have introduced before. They linearly combine several (a priori unfair) basis predictors, under a post-processing perspective. We will discuss several penalised criteria to find a convenient trade-off between the predictive power of the new predictors and their fairness.

4.1. The Idea of Model Averaging

To fix the ideas, assume we have $M$ predictors of $Y$ from $X$ at our disposal. When $Y$ is discrete and binary, such classifiers will be denoted $g_{j}$ , $j \in {1, \dots, M}$ : any $g_{j} (x)$ yields an approximated value of $P (Y = 1 | X = x)$ . Note that $g_{j} (x)$ may be binary $0$ - $1$ itself, but it is not mandatory. When $Y$ is continuous, our predictors are now regressors and are denoted as $r_{j}$ , $j \in {1, \dots, M}$ : any $r_{j} (x)$ yields an approximated value of $E (Y | X = x)$ . In the binary case, it is common to define binary classifiers $g_{j}$ from an underlying (continuous) regressor $r_{j}$ simply by setting $g_{j} (x) = 1 (r_{j} (x) > 1 / 2)$ .⁴

An ensemble of $M$ decision trees (called additive tree model or ATM) constitutes an important particular case of bagging. Let $X \in X$ be the vector of individual features. Let $g_{j} (x)$ be the output obtained from tree $j$ , $j \in {1, \dots, M}$ . For classification purpose, the output $G$ of the additive tree model is a weighted sum of all the tree outputs, i.e., $G (x) = \sum_{j = 1}^{M} ω_{j} g_{j} (x)$ , where $ω_{j} \in R$ is the weight associated with tree $j$ . Then, in the case of binary outputs, fix a threshold $a$ so that $g (x) = 1 (G (x) \geq a)$ . Most often in machine learning, $g_{j} (x) \in [0, 1]$ , $ω_{j} = 1 / M$ for every $j$ and $a = 1 / 2$ .

In the case of regressions, the idea of bagging (Breiman, 1996) is similar: combine $M$ regression-type predictors $r_{j}$ to build another predictor $r_{ω} (x) = \sum_{j = 1}^{M} ω_{j} r_{j} (x) =: ω^{'} r (x)$ for every $x \in X$ , denoting $r (x) := (r_{1} (x), \dots, r_{M} (x))$ . For the properties on bagging estimates, we refer to Breiman (2001), Friedman and Hall (2007), among others. The vector of weights $ω := (ω_{1}, \dots, ω_{M})$ has to be carefully chosen. Even if $ω$ could depend on $x$ , we impose it is a constant vector for the sake of simplicity. In the usual case of random forests, they are all equal even if a few attempts have been made to add another degree of flexibility with different weights (Kim et al., 2011; Maudes et al., 2012 e.g.). Here, we only impose $\sum_{j = 1}^{M} ω_{j} = ω^{'} e = 1$ , but we do not restrict ourselves to nonnegative weights. Therefore, $ω$ may be any arbitrary vector in $R^{M}$ so that $ω^{'} e = 1$ .

The previous formulation is very general and includes some popular models as special cases, like random forests. The latter additive tree model is widely used in real-world applications and appears as the most popular and powerful off-the-shelf classifer (Breiman, 2001). It can cope with regression and multi-class classification for categorical or numerical variables with high degree of accuracy. In essence, random forest is a bagging model of trees where each tree is trained on a group of randomly sampled instances with randomly selected features.⁵

Now, we would like to calibrate $ω$ to obtain a predictor $r_{ω}$ that fulfils “at most” some of the latter concepts of fairness. In other words, is it possible to linearly combine several possibly unfair predictors so that the final predictor becomes fair? We will see below that perfect fairnes can sometimes be obtained, and approximated fairness can always be targeted. For instance, in the case of demographic parity and equalised odds, the weights of this “fair bagging” predictor will be chosen so that some moment conditions as (3.3) or (3.5) respectively will be empirically fulfilled. Note that such conditions are necessary but in general not sufficient to insure fairness.

Typically, as for random forests, the number $M$ of basis classifiers/trees may be large (several hundreds, for instance). This is not a requirement for fairness purpose in our framework. Indeed, we do not need many competing predictors to build approximately fair new ones, as theoretically proved in this section and illustrated in Section 5. Only two or three classifiers/regressors can do the task, particularly when they yield significantly different forecasts: a mixture of a few different predictors will be better than mixing a large number of “closely related” basis models, particularly when they are related to the same family of models.

Until now, we have not discussed the predictive power of such new models by model averaging. In other words, it is likely that an “optimal” weighting scheme that only seeks to improve fairness will deteriorate its performances in terms of prediction, compared to the initial classifiers/regressors (or at least the best ones). Intuitively, when we use fair predictors based on weighted averages of some “reasonably performing” basis predictors, the new performances should be “average”, in a rough sense. Specifically, they should belong to the range of the performances obtained with the initial predictors, at least when all weights are nonnegative and sum to one. We will propose to directly manage the trade-off between fairness and predictive power by considering penalised criteria. They would simultaneously take into account the loss due to unfairness and the loss due to bad predictions. The share between the two losses will be tackled through a tuning parameter $λ \geq 0$ .

4.2. Penalized Criteria

For every fairness concept, our aim is to find an optimal vector of weights $ω \in Δ$ , for some convex subset $Δ \in R^{M}$ , that will minimise a penalised criterion. To be specific, we would like to evaluate

ω^{*} := \arg min_{ω \in Δ} {Q_{1} (ω) + λ Q_{2} (ω)},

(4.1)

for some given nonnegative tuning parameter

λ

, where

Q_{1} (ω)

(resp.

Q_{2} (ω)

) measures the unfairness (resp. lack of accuracy) of the predictor obtained using the vector of weights

ω

. The criteria

Q_{k}

are theoretical quantities that will be written as expectations, mainly for convenience. In practise, they are unknown and have to be estimated by some empirical counterparts

Q_{n, k} (ω)

k \in {1, 2}

. The estimated “optimal” vector of weights will then be defined by

{\hat{ω}}^{*} := \arg min_{ω \in Δ} {Q_{n, 1} (ω) + λ Q_{n, 2} (ω)} .

(4.2)

Clearly, the quantities

{\hat{ω}}^{*}

can be seen as extremum estimators of

ω^{*}

. Their consistency or even their asymptotic normality could be proven (most often easily) under some conditions of regularity: see Theorem 2.1 and 3.1 in Newey and McFadden (1994), for instance.⁶

Note that solving (4.1) would not yield strictly speaking a fair predictor by averaging simpler predictors. Our logic is rather to exhibit predictors that are “approximately fair”, the result of a reasonable trade-off between fairness and accuracy. Many authors already agreed that imposing strict fairness is often a too strong requirement, or at least too costly in terms of prediction power. See the penalised approach of Centorrino et al. (2022), the $ϵ$ -fairness concept in Donini et al. (2018), the maximisation of accuracy under some fairness bounds in Zafar et al. (2017), the reduction approach of Agarwal et al. (2018), among others.

By considering linear combinations of basis predictors, our model encompasses all these simpler models. In other words, model averaging enriches the set of the initial models, offering more flexibility. As a consequence, after convenient calibration, we can achieve better fairness that individual predictors, and similarly in terms of accuracy. Concerning the two latter dimensions simultaneously, we exhibit a set of “optimal models” by playing with the hyperparameter $λ$ . The latter family of models is Pareto-optimal, in the sense our solutions cannot be strictly dominated by any basis predictor in terms of fairness and accuracy at the same time. See Guardieiro et al. (2023) or Di Carlo et al. (2023) for similar ideas.

To approximately satisfy some moment conditions, we rely on an i.i.d. sample $S_{n} := (X_{i}, Y_{i})_{i = 1, \dots, n}$ , $X_{i} = (S_{i}, Z_{i})$ . Generally speaking, the quality of a predictor is measured by a comparison between predicted values - here $ω^{'} r (X_{i})$ - and the true realizations $Y_{i}$ , that may be discrete or continuous. Let $d$ be a distance between two real numbers, typically $d_{1} (x, y) := | x - y |$ or $d_{2} (x, y) := (x - y)^{2}$ .

(a)

Let us first deal with demographic parity, as defined by the constraint (3.3). A natural criterion of fairness is here $Q_{1} (ω) := | E [ω^{'} r (X)] - E [ω^{'} r (X) | S = 0] |^{a}$ , for some real number $a > 0$ . The quality of the fit will be evaluated by $Q_{2} (ω) := E [d (ω^{'} r (X), Y)]$ .

The empirical versions of the latter constraints are obtained by replacing theoretical expectations by empirical ones, based on the calibration subset. A convenient choice of $ω$ should minimise a distance between $\sum_{i = 1}^{n} 1 (S_{i} = 0) ω^{'} r (X_{i}) / \sum_{i = 1}^{n} 1 (S_{i} = 0)$ and $\sum_{i = 1}^{n} ω^{'} r (X_{i}) / n$ , that are well-known consistent estimators of $E [ω^{'} r (X_{i}) | S = 0]$ and $E [ω^{'} r (X_{i})]$ respectively. Thus, in the case of demographic parity, an intuitive penalised criterion for finding optimal weights would be

\arg min_{ω} | \frac{\sum_{i = 1}^{n} 1 (S_{i} = 0) ω^{'} r (X_{i})}{\sum_{i = 1}^{n} 1 (S_{i} = 0)} - \frac{1}{n} \sum_{i = 1}^{n} ω^{'} r (X_{i}) |^{a} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i}),

(4.3)

under the constraints

ω^{'} e = 1

, and possibly

ω_{k} \in [0, 1]

k \in {1, \dots, M}

. We promote to set

a = 2

, even if it the latter choice is largely arbitrary. Indeed, with

a = 2

, the corresponding fairness criterion becomes a smooth (differentiable) map of

ω

and, more interestingly, some optimal vectors of weights can be obtained in closed-form (see below). Note that, if

d = d_{2}

, the latter programme can be rewritten

\arg min_{ω} {ω^{'} Δ_{D P} Δ_{D P}^{'} ω + λ ω^{'} \bar{Σ} ω},

(4.4)

by setting

Δ_{D P} := \frac{\sum_{i = 1}^{n} 1 (S_{i} = 0) r (X_{i})}{\sum_{i = 1}^{n} 1 (S_{i} = 0)} - \frac{1}{n} \sum_{i = 1}^{n} r (X_{i}), \bar{Σ} := \frac{1}{n} \sum_{i = 1}^{n} (r (X_{i}) - Y_{i} e) (r (X_{i}) - Y_{i} e)^{'} .

Obviously, we denote by

e

the

M

-vector whose components are one. It clearly appears that (4.4) is formally the same mathematical problem as for a minimum-variance portfolio optimisation programme. Without imposing

ω_{k} \in [0, 1]

, it is well-known it has a unique solution

{\hat{ω}}_{D P}^{*} := Σ_{D P}^{- 1} e / e^{'} Σ_{D P}^{- 1} e

, with

Σ_{D P} := Δ_{D P} Δ_{D P}^{'} + λ \bar{Σ}

. By choosing the tuning parameter

λ

, we manage a trade-off between targeting perfect fairness (

λ = 0

) and picking-up the “best” linear combination of basis predictors in terms of forecasting performance (

λ >> 1

). Remark 4.1

When $λ = 0$ , Programme (4.3) reduces to

\arg min_{ω} {(\sum_{i = 1}^{n} 1 (S_{i} = 0) ω^{'} r (X_{i}) / \sum_{i = 1}^{n} 1 (S_{i} = 0) - \sum_{i = 1}^{n} ω^{'} r (X_{i}) / n)}^{2},

(4.5)

under the constraints

ω^{'} e = 1

and (possibly)

ω_{k} \in [0, 1]

for every

k = 1, \dots, M

. The latter optimisation programme can be easily solved by quadratic programming. Sometimes, it yields degenerate solutions for which all the weights are zero, except for a single component. In general, there are an infinite number of solutions of (4.5). Indeed, there exist an infinity of vectors

ω

s.t.

ω^{'} e = 1

and

\sum_{i = 1}^{n} 1 (S_{i} = 0) ω^{'} r (X_{i}) / \sum_{i = 1}^{n} 1 (S_{i} = 0) = \sum_{i = 1}^{n} ω^{'} r (X_{i}) / n,

strictly speaking, given by the intersection of two hyperplanes in

R^{M}

. Therefore, to obtain a unique solution

ω

, it is necessary to select a tuning parameter

λ > 0

Remark 4.2

Alternatively, assume there are an infinity of solutions ${ω | ω^{'} v_{n} = 0}$ of (4.5) with $λ = 0$ , for some $M$ -vector $v_{n}$ . When $ω \in [0, 1]^{M}$ is not imposed, this is the case almost everywhere. A natural idea would then be to select the best “fair predictor” among the latter ones, namely among the “perfectly fair” predictors. This can be done by solving the programme

\arg min_{ω} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i}), under ω^{'} e = 1 and ω^{'} v_{n} = 0.

(4.6)

The latter programme has always a solution

ω \in R^{M}

, for “reasonable” distances

d

such as

d = d_{2}

. If we impose

ω \in [0, 1]^{M}

in addition, (4.6) may have no solution, but only when all the components of

v_{n}

have the same sign. Under such circumstances, a practical solution would be to try different basis predictors until the vector

v_{n}

has some components of different signs, or to (slightly) disturb one of the existing basis predictors. The advantage of the latter method is twice: we will be sure that our predictor by model averaging will be fair, and no choice of

λ

is necessary. Nonetheless, by reducing the family of candidates

ω

in this way, there is a high risk of restricting ourselves to poorly predictive “bagging” predictors.

(b)

Concerning equalised odds, the constraint (3.5) has to be satisfied for every $y^{'}$ , yielding the loss function

Q_{1} (ω) = E_{Y} [| E_{X} [ω^{'} r (X) | Y] - E_{X} [ω^{'} r (X) | S = 0, Y] |^{a}], a > 0.

The latter criterion may be empirically approximated. To this aim, we introduce some nonparametric estimators of

E [\hat{Y} | S = k, Y = y]

k \in {0, 1}

, and of

E [\hat{Y} | Y = y]

. For analytical simplicity, we choose the usual Nadaraya-Watson estimators: for every

k \in {0, 1}

, set

E [\hat{Y} | S = k, Y = y] ≃ {\hat{r}}_{k} (y) := \frac{\sum_{i = 1}^{n} ω^{'} r (X_{i}) 1 (S_{i} = k) K_{h} (Y_{i} - y)}{\sum_{i = 1}^{n} 1 (S_{i} = k) K_{h} (Y_{i} - y)} =: ω^{'} v_{k, h} (y),

for some univariate kernel

K

, a bandwidth sequence

h = h (n) \to 0

and

K_{h} (t) := K (t / h) / h

for every

t

. The kernel

K

is simply a measurable map

K : R \to R

s.t.

\int K = 1

. Similarly, we have

E [\hat{Y} | Y = y] ≃ \hat{r} (y) := \frac{\sum_{i = 1}^{n} ω^{'} r (X_{i}) K_{h} (Y_{i} - y)}{\sum_{i = 1}^{n} K_{h} (Y_{i} - y)} =: ω^{'} v_{h} (y) .

Note that

{\hat{r}}_{k} (y)

and

\hat{r} (y)

are linear function of

ω

, for any

y

. Thus, the EO programme to be solved will be

\arg min_{ω} \frac{1}{n} \sum_{i = 1}^{n} | {\hat{r}}_{0} (Y_{i}) - \hat{r} (Y_{i}) |^{a} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i}),

(4.7)

for some constant

a > 0

, under the constraint

ω^{'} e = 1

. When

d = d_{2}

and

a = 2

, this means solving

\begin{aligned} \arg min_{ω} {ω^{'} Σ_{h} ω + \frac{λ}{n} \sum_{i = 1}^{n} (ω^{'} r (X_{i}) - Y_{i})^{2}} = {ω^{'} Σ_{h} ω + λ ω^{'} \bar{Σ} ω}, where \end{aligned}

\begin{aligned} Σ_{E O} := Σ_{h} + λ \bar{Σ}, Σ_{h} = n^{- 1} \sum_{i = 1}^{n} (v_{0, h} (Y_{i}) - v_{h} (Y_{i})) (v_{0, h} (Y_{i}) - v_{h} (Y_{i}))^{'} . \end{aligned}

Its unique solution in

R^{M}

{\hat{ω}}_{E O}^{*} := Σ_{E O}^{- 1} e / e^{'} Σ_{E O}^{- 1} e

. Obviously, it is possible to add the constraints

ω_{k} \in [0, 1]

for every

k

. As a practical consequence, this implies losing analytic solutions and relying on numerical optimisation (quadratic programming). Remark 4.3

Under some conditions of regularity, it can be shown that $Σ_{h}$ is a consistent estimator of

Σ := E_{Y} [(E_{X} [r (X) | Y] - E_{X} [r (X) | Y, S = 0]) (E_{X} [r (X) | Y] - E_{X} [r (X) | Y, S = 0])^{'}],

when

n \to \infty

and

h \to 0

. Unfortunately, when

M >> 1

as in the case of random forests, the covariance matrix

Σ

may be high dimensional, and the sample covariance matrix

Σ_{h}

could suffer from a significant amount of sampling error. Its inverse, that is required when

λ = 0

, may become a poor estimator of

Σ^{- 1}

or it may not even exist. In particular, when

Y

is discrete and can take

p

values, it is easy to check that the rank of

Σ_{h}

is at most

min (p, M)

. Thus, for a binary

Y

, the latter matrix is no longer invertible when we consider a linear combination of three or more basis classifiers ! Fortunately, when

λ > 0

, the matrix

Σ_{E O}

is in general invertible. Note the natural link with shrinkage methods.⁷

(c)

Under the point of view of LDM, let us still choose a $L^{a}$ distance for some $a > 0$ , and set

Q_{1} (ω) := | E [| ω^{'} r (X) - Y |^{a} | S = 1] - E [| ω^{'} r (X) - Y |^{a} | S = 0] |^{b},

for some constant

b > 0.

An estimator of

E [| \hat{Y} - Y |^{a} | S = k]

k \in {0, 1}

, is given by

E [| \hat{Y} - Y |^{a} | S = k] ≃ \frac{\sum_{i = 1}^{n} | ω^{'} r (X_{i}) - Y_{i} ω^{'} e |^{a} 1 (S_{i} = k)}{\sum_{i = 1}^{n} 1 (S_{i} = k)} =: {\hat{E}}_{a, k} (ω) .

A general programme for lack of disparate mistreatment would be

\arg min_{ω} {| {\hat{E}}_{a, 0} (ω) - {\hat{E}}_{a, 1} (ω) |^{b} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i})} .

Setting

a = 2

, this yields

\begin{aligned} E [(\hat{Y} - Y)^{2} | S = k] & ≃ \frac{\sum_{i = 1}^{n} (ω^{'} r (X_{i}) - Y_{i} ω^{'} e)^{2} 1 (S_{i} = k)}{\sum_{i = 1}^{n} 1 (S_{i} = k)} \\ = ω^{'} \frac{\sum_{i = 1}^{n} (r (X_{i}) - Y_{i} e) (r (X_{i}) - Y_{i} e)^{'} 1 (S_{i} = k)}{\sum_{i = 1}^{n} 1 (S_{i} = k)} ω =: ω^{'} {\bar{Σ}}_{k} ω, \end{aligned}

under the constraint

ω^{'} e = 1

and with obvious notations. Note that

E [(\hat{Y} - Y)^{2}] ≃ ω^{'} \bar{Σ} ω,

and that

\bar{Σ}

is a convex combination of

{\bar{Σ}}_{0}

and

{\bar{Σ}}_{1}

. Therefore, when

a = 2

, lack of disparate mistreatment implies solving

\arg min_{ω} {| ω^{'} ({\bar{Σ}}_{0} - {\bar{Σ}}_{1}) ω |^{b} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i})},

(4.8)

under the constraint

ω^{'} e = 1

, and possibly under

ω_{k} \in [0, 1]

for every

k \in {1, \dots, M}

. When

d = d_{2}

a = b = 2

and without the latter constraints, the solution of (4.8) is

{\hat{ω}}_{L D M}^{*} := Σ_{L D M, γ}^{- 1} e / e^{'} Σ_{L D M, γ}^{- 1} e,

(4.9)

where

Σ_{L D M, γ} := 2 γ ({\bar{Σ}}_{0} - {\bar{Σ}}_{1}) + λ \bar{Σ},

for some positive constant

γ

that is the root of the nonlinear equation

γ = ({\hat{ω}}_{L D M}^{*})^{'} ({\bar{Σ}}_{0} - {\bar{Σ}}_{1}) {\hat{ω}}_{L D M}^{*}

. See Section B in the appendix. Remark 4.4

To obtain “perfect fairness’, the goal would be to find a vector $ω$ so that $ω^{'} ({\bar{Σ}}_{0} - {\bar{Σ}}_{1}) ω = 0$ , and $ω^{'} e = 1$ . The latter programme has no solution if the symmetric matrix ${\bar{Σ}}_{0} - {\bar{Σ}}_{1}$ is strictly positive or strictly negative. Otherwise, there are an infinite number of solutions (the isotropic cone). Alternatively, a unique $ω$ and perfect LDM are reachable by adapting Remark 4.2 in the LDM case.

(d)

Concerning conditional fairness given a vector of covariates, the picture is more complex because we have to manage multiple conditioning random vectors. Depending on our definition of fairness (DP, EO or LDM), denote by $W$ the relevant conditioning vector. With the notations of Section 3 and to obtain conditional demographic parity (or LDM) given $Z_{1}$ , define $W = Z_{1}$ and in the case of conditional equalised odds given $Z_{1}$ , set $W = (Z_{1}, Y)$ .

To build a fair classifier/regressor by linear combinations of basis predictors in the DP or EO cases, we would like to fulfil some moment-type relationships as

E [ω^{'} r (W, 0) | S = 0, W = w] = E [ω^{'} r (W, 1) | S = 1, W = w]

“at best” for all values of

w

. There are many ways of enforcing such constraints, at least approximately. As for chi-squared test statistics, the simplest approach is to discretise the conditioning variable

W

: build

m

buckets of

W

values, denoted as

W_{1}, \dots, W_{m}

. The corresponding subsamples of

S_{n}

are denoted

S_{n, 1}, \dots, S_{n, m}

S_{n} = \cup_{k = 1}^{m} S_{n, k}

Card S_{n, k} =: n_{k}

. Any previously defined criteria written as

\arg min_{ω} L_{n} (ω, S_{n})

has a conditional version

\arg min_{ω} \sum_{k = 1}^{m} π_{k} L_{n_{k}} (ω, S_{n, k})

\sum_{k} π_{k} = 1

. A natural choice would be

π_{k} := n_{k} / n

for every

k

An alternative direct approach is possible, that does not necessitate partitioning the $W$ subspace. Indeed, as in Section 3, define an estimator of $E [r (X) | S = k, W = w]$ by

r (w, k) := \frac{\sum_{i = 1}^{n} r (X_{i}) 1 (S_{i} = k) K_{h} (W_{i} - w)}{\sum_{i = 1}^{n} 1 (S_{i} = k) K_{h} (W_{i} - w)}, k \in {0, 1} .

Now, since

W

may be multivariate and is of dimension

s

, the latter kernel

K

denotes a measurable map from

R^{s}

R

s.t.

\int K (x) d x = 1

. For simplicity, consider a vector of bandwidths

h = (h_{1}, \dots, h_{s})

and denote

K_{h} (x) := K (x_{1} / h_{1}, \dots, x_{s} / h_{s})

for any

s

-dimensional vector

x = (x_{1}, \dots, x_{s})

. Therefore, in the case of conditional DP and EO, a solution could be

{\hat{ω}}_{c f} := \arg min_{ω} {\frac{1}{n} \sum_{i = 1}^{n} | ω^{'} r (W_{i}, 0) - ω^{'} r (W_{i}, 1) |^{a} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i})},

(4.10)

under the constraint

ω^{'} e = 1

and for some

a > 0

. Obviously,

W_{i} = Z_{1, i}

for conditional DP,

W_{i} = (Z_{1, i}, Y_{i})

for conditional EO, and the theoretical fairness criterion is

Q_{1} (ω) = E_{W} [| E_{X} [ω^{'} r (X) | W] - E_{X} [ω^{'} r (X) | S = 0, W] |^{a}] .

As above and when

a = 2

, the solution of (4.10) is

{\hat{ω}}_{c f} = Σ_{c f}^{- 1} e / e^{'} Σ_{c f}^{- 1} e,

where

Σ_{c f} := n^{- 1} \sum_{i = 1}^{n} (r (W_{i}, 0) - r (W_{i}, 1)) (r (W_{i}, 0) - r (W_{i}, 1))^{'} + λ \bar{Σ} .

In the same spirit, concerning conditional lack of disparate mistreatment with a quadratic loss, the goal is to find a vector of weights

ω

s.t.

E [(ω^{'} r (W, 0) - Y)^{2} | S = 0, Z_{1} = z] = E [(ω^{'} r (W, 1) - Y)^{2} | S = 1, Z_{1} = z],

for every

z

. Note that the latter identity involves a quadratic form of

ω

, but indexed by

z

. For

k \in {0, 1}

and with obvious notations, define

{\bar{Σ}}_{k} (z) := \frac{\sum_{i = 1}^{n} (r (X_{i}) - Y_{i} e) (r (X_{i}) - Y_{i} e)^{'} 1 (S_{i} = k) K_{h} (Z_{i} - z)}{\sum_{i = 1}^{n} 1 (S_{i} = k) K_{h} (Z_{i} - z)} \cdot

For conditional LDM, we then propose to solve

{\hat{ω}}_{c l d m} (z) := \arg min_{ω} n^{- 1} \sum_{i = 1}^{n} {ω^{'} ({\bar{Σ}}_{0} (Z_{1, i}) - {\bar{Σ}}_{1} (Z_{1, i})) ω}^{2} + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i}),

under the constraint

ω^{'} e = 1

and, possibly,

ω_{j} \in [0, 1]

for every

j

The weights associated to our basis selectors intuitively behave as follows: if one of the basis selectors is better than the others both in terms of accuracy and fairness, it will capture most of the total weight (the $L^{1}$ norm of $ω$ ). If it is better in terms of accuracy but not in terms of fairness (or the opposite), it will share the total weight of $ω$ with at least another basis selector. The optimal values of the components $ω_{k}$ depend on $λ$ and on the relative performances of these selectors in the two considered dimensions. When $ω$ is left unconstrained, some negative values $ω_{k}$ can be observed. Even if this phenomenon is not very intuitive, it can be understood as a way of penalizing some predictors that provide particularly inaccurate and/or unfair forecasts for some subsets of $X$ values, and overweighting the best suited predictors for the opposite reason.

Even if most papers in the literature focus on binary outcomes (classification), we stress that our methodology through penalised criteria obviously applies to general regression-type problems, as illustrated in Section 5.3. In other words, $Y$ and their predictors $r_{j} (x)$ , $j \in {1, \dots, M}$ , may be arbitrary real numbers (not only binary variables in ${0, 1}$ ).

4.3. Discretization Issues

In the case of binary explained variables $Y \in {0, 1}$ , it may seem strange to calibrate our weights $ω$ , or, even more, to evaluate the quality of our new classifier by comparing continuous outcomes $ω^{'} r (X_{i})$ to binary observations $Y_{i}$ . In other words, it is tempting to replace the predictor $p_{i, 0} (ω) := ω^{'} r (X_{i})$ with $p_{i, 1} (ω) := 1 (ω^{'} r (X_{i}) > 0.5),$ for every $i$ . Therefore, this opens the door to several slightly different calibration criteria. When no closed form solutions are required, potentially all of them are meaningful. Let us propose a general framework that encompasses all situations.

(a)
In the case of demographic parity, a general penalised criterion for finding optimal weights is
$\arg min_{ω} {| \frac{\sum_{i = 1}^{n} 1 (S_{i} = 0) p_{i, k} (ω)}{\sum_{i = 1}^{n} 1 (S_{i} = 0)} - \frac{\sum_{i = 1}^{n} 1 (S_{i} = 1) p_{i, k} (ω)}{\sum_{i = 1}^{n} 1 (S_{i} = 1)} |}^{a} + \frac{λ}{n} \sum_{i = 1}^{n} d (p_{i, l} (ω), Y_{i}),$
(4.11)
for some $a > 0$ and every choice of $(k, l) \in {0, 1}^{2}$ , under the constraint $ω^{'} e = 1$ (and possibly $ω_{j} \in [0, 1]$ , $j \in {1, \dots, M}$ ).
(b)
In the case of equalised odds, it is necessary to redefine some quantities of interest: for every $(k, s) \in {0, 1}^{2}$ . We have
$E [p_{i, k} (ω) | S = s, Y = y] ≃ q_{k} (s, y, ω) := \frac{\sum_{i = 1}^{n} p_{i, k} (ω) 1 (S_{i} = s) K_{h} (Y_{i} - y)}{\sum_{i = 1}^{n} 1 (S_{i} = s) K_{h} (Y_{i} - y)} \cdot$
When $Y$ is binary, simply choose $K (t) = 1 (t \in [- 0.5, 0.5])$ and any bandwidth $h \in (0, 1)$ , for instance. Then, the latter estimator becomes
$q_{k} (s, y, ω) := \frac{\sum_{i = 1}^{n} p_{i, k} (ω) 1 (S_{i} = s, Y_{i} = y)}{\sum_{i = 1}^{n} 1 (S_{i} = s, Y_{i} = y)} \cdot$
When $k = 0$ , the map $ω \mapsto q_{k} (s, y, ω)$ yields a smooth consistent estimator of $E [\hat{Y} | S = s, Y = y]$ . When $k = 1$ , this map becomes a non differentiable function of $ω$ .

The generalised penalised-EO programme could be defined as
$\arg min_{ω} \frac{1}{n} \sum_{i = 1}^{n} | q_{k} (0, Y_{i}, ω) - q_{k} (1, Y_{i}, ω) |^{a} + \frac{λ}{n} \sum_{i = 1}^{n} d (p_{i, l} (ω), Y_{i}),$
(4.12)
for some $a > 0$ and any couple $(k, l) \in {0, 1}^{2}$ , under the constraint $ω^{'} e = 1$ (and possibly $ω_{j} \in [0, 1]$ , $j \in {1, \dots, M}$ ).
(c)
Concerning lack of disparate mistreatment, for every $s$ and $k$ in ${0, 1}$ , note that
$E [| \hat{Y} - Y |^{a} | S = s] ≃ \frac{\sum_{i = 1}^{n} | p_{i, k} (ω) - Y_{i} ω^{'} e |^{a} 1 (S_{i} = s)}{\sum_{i = 1}^{n} 1 (S_{i} = s)} =: ℓ_{k, s} (ω), a > 0.$
This leads to new generalised LDM criteria and programmes as
$\arg min_{ω} | ℓ_{k, 0} (ω) - ℓ_{k, 1} (ω) |^{b} + \frac{λ}{n} \sum_{i = 1}^{n} d (p_{i, l} (ω), Y_{i}), b > 0,$
(4.13)
for every choice of $(k, l) \in {0, 1}^{2}$ , under the constraints $ω^{'} e = 1$ .
When $(k, l) = (1, 1)$ , the latter programmes involve non-differentiable functions of $ω$ and there is no unique solution, beside numerical burden. To enforce a single solution, we use hereafter a numerical trick: replace the step functions $ω \mapsto 1 (ω^{'} r (X_{i}) > 0.5)$ with the differentiable map $ω \mapsto Φ ((ω^{'} r (X_{i}) - 0.5) / σ)$ , for some (small) constant $σ > 0$ . Here, $Φ$ denotes the cdf of a $N (0, 1)$ .
4.4. Extension to Multiple Criteria of Fairness

Interestingly, note that our penalised loss framework can easily take into account several definitions of fairness simultaneously, through slightly more complex programmes. For instance, if a user hesitates between Demographic Parity and Equalized Odds, the best suited solution by model averaging would be to mix (4.3) and (4.7). When $a = 2$ , this means solving the programme

\begin{aligned} \arg min_{ω} {(\frac{\sum_{i = 1}^{n} 1 (S_{i} = 0) ω^{'} r (X_{i})}{\sum_{i = 1}^{n} 1 (S_{i} = 0)} - \frac{1}{n} \sum_{i = 1}^{n} ω^{'} r (X_{i}))}^{2} + \frac{μ}{n} \sum_{i = 1}^{n} ({\hat{r}}_{0} (Y_{i}) - \hat{r} (Y_{i}))^{2} \\ + \frac{λ}{n} \sum_{i = 1}^{n} d (ω^{'} r (X_{i}), Y_{i}), \end{aligned}

for two nonnegative tuning parameters

λ

and

μ

More generally, taking into account all fairness criteria and all potential discretization options, a “meta-program” to simultaneously manage DP, EO and LDM would be a synthesis of (4.11), (4.12) and (4.13). With the notations of Section 4.3, this means

\begin{aligned} \arg min_{ω} μ_{0} {| \frac{\sum_{i = 1}^{n} 1 (S_{i} = 0) p_{i, k_{0}} (ω)}{\sum_{i = 1}^{n} 1 (S_{i} = 0)} - \frac{\sum_{i = 1}^{n} 1 (S_{i} = 1) p_{i, k_{0}} (ω)}{\sum_{i = 1}^{n} 1 (S_{i} = 1)} |}^{a_{0}} \\ + \frac{μ_{1}}{n} \sum_{i = 1}^{n} | q_{k_{1}} (0, Y_{i}, ω) - q_{k_{1}} (1, Y_{i}, ω) |^{a_{1}} + μ_{2} | ℓ_{k_{2}, 0} (ω) - ℓ_{k_{2}, 1} (ω) |^{b} + \frac{λ}{n} \sum_{i = 1}^{n} d (p_{i, l} (ω), Y_{i}), \end{aligned}

for every choice of

(k_{0}, k_{1}, k_{2}, l) \in {0, 1}^{4}

(μ_{0}, μ_{1}, μ_{2}) \in R_{+}^{3}

, some positive numbers

a_{0}

a_{1}

a

(in the definition of the

ℓ_{k, s}

maps) and

b

, and under the constraint

ω^{'} e = 1

(possibly with

ω_{k} \in [0, 1]

k \in {1, \dots, M}

Numerous definitions of fairness are available in the literature. In this paper, we have focussed on four of them, as defined in Section 3. It would be easy to adapt our ideas and our penalised criteria to other concepts of fairness, once they can be rewritten as equalities between some conditional expectations. Sometimes, none of these concepts is self-evident and it is not obvious to discriminate between several criteria of fairness. For instance, choosing between “separation” ( $\hat{Y}$ is independent of $S$ given $Y$ ) and “sufficiency” ( $Y$ is independent of $S$ given $\hat{Y}$ ), as defined in Barocas et al. (2023), may be questionable. Moreover, some definitions of fairness may even be incompatible. Thus, in the case of binary classifiers, either demographic parity holds or equalised odds but not both in general (Prop. 2.1 in del Barrio et al. 2020). See Berk et al. (2018), Kleinberg et al. (2017). It is therefore important to note that our methodology makes it possible to improve fairness under several concepts of fairness simultaneously. As written in Makhlouf et al. (2021), “with the large number of fairness notions and the subtle resemblance between Machine Learning-based decision-making scenarios, deciding about which fairness notion to use is not a trivial task.” In other words, we have proposed a solution to a significant challenge in fairness research: the difficulty of satisfying multiple definitions simultaneously due to tensions and conflicts between different notions of fairness (Barocas et al., 2023, Chapter 3).

5. Empirics

In this section, we illustrate our way of improving the fairness of some predictive models, and also the difficulty of this task when a high predictive power is required. This trade-off is well known in the literature. We will stress the practical importance of the tuning parameter $λ$ we introduced before. First, let us define the performance measures that will be used for each concept of fairness.

5.1. Performance Measures

Consider general binary and/or continuous predictive models $\hat{Y} = g (X)$ , and $X = (Z_{1}, Z_{2}, S)$ with the same notations as above: $S \in {0, 1}$ is the sensitive feature; $Z_{1} \in R^{q}$ is a potential conditional feature of interest for testing conditional fairness; and $Z_{2} \in R^{p - q}$ is stacking the other underlying explanatory variables. Our “corrected” predictor will be chosen among those proposed in Section 4, i.e., given by (4.11), (4.12) and (4.13).

Many measures of fairness may be chosen, depending on the concepts we consider (Friedler et al., 2019, e.g.). We will work with the following ones, that are relevant for any continuous or binary explained variable $Y$ , keeping in mind they largely remain arbitrarily in the absence of consensus in the literature:

in the case of demographic parity, the “disparate impact” is defined as

D I_{α} := E [| \hat{Y} |^{α} | S = 0] / E [| \hat{Y} |^{α} | S = 1], for some α > 0.

When

α = 1

and

\hat{Y}

is nonnegative, as for most classifiers, we recover the early notion of disparate impact (Feldman et al., 2015).

in the case of equalised odds, we propose the measure of “mean $L^{β}$ fairness”

M L F_{β} := 1 - E_{Y} [| E (\hat{Y} | S = 0, Y) - E (\hat{Y} | S = 1, Y) |^{β}], for some β > 0;

in the case of lack of disparate mistreatment, define the “disparate mistreatment rate” as

D M R_{γ} := E [| \hat{Y} - Y |^{γ} | S = 0] / E [| \hat{Y} - Y |^{γ} | S = 1], for some γ > 0.

In the previous definitions, we recall that expectations have to be understood w.r.t.

(Y, X)

and “out-of-sample”. In other words, the way our predictor

\hat{Y}

has been calibrated/estimated has not to be taken into account in the evaluation of such measures of fairness. The latter map is simply managed as a deterministic function. Note that the latter measures may not belong to

[0, 1]

in general, even if they are equal to one in the case of perfect fairness.

To measure conditional fairness, the same concepts can be easily adapted. For instance, a measure of conditional demographic parity given $Z_{1} = z$ is simply $D I_{α} (z) := E [| \hat{Y} |^{α} | S = 0, Z_{1} = z] / E [| \hat{Y} |^{α} | S = 1, Z_{1} = z]$ . And similarly for $M L F_{β}$ and $D M R_{γ}$ .

To evaluate the accuracy of our predictors, we simply rely on the classical mean-squared error

M S E := E [(\hat{Y} - Y)^{2}] = E [(g (X) - Y)^{2}] .

The fairness measures mentioned above will be empirically estimated, based on some test dataset that does not intersect the sample used for calibration purpose. To be more specific, and as shown in Figure 1 we first use the train dataset to estimate our basis estimators.

Figure 1.

The data split process.

This stage remained hidden in Section 4, because we considered some “given” basis estimators. Then, based on the calibration dataset $S_{n} := {(X_{i}, Y_{i}), i = 1, \dots, n}$ , we estimate our vectors of weights $ω$ by the optimisation processes introduced before. At last, with such optimal weights and the predictions given by the combined estimators on the test dataset, we can evaluate fairness and prediction performances. Denote $T_{\bar{n}} := {({\tilde{X}}_{i}, {\tilde{Y}}_{i}), i = 1, \dots, \bar{n}}$ the test dataset. Obviously, we suppose that the observations in $T_{\bar{n}}$ follow the same distribution as in the calibration set $S_{n}$ . The empirical counterparts of our previously defined fairness measures are defined as follows. (a)

Demographic parity: the empirical $α$ -disparate impact is

{\hat{D I}}_{α} := \frac{\sum_{i = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) |^{α} 1 ({\tilde{S}}_{i} = 0) / \sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = 0)}{\sum_{i = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) |^{α} 1 ({\tilde{S}}_{i} = 1) / \sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = 1)}, α > 0.

(b)

Equalized odds: the empirical mean $L^{β}$ fairness is ${\hat{M L F}}_{β} := 1 - \frac{1}{\bar{n}} \sum_{j = 1}^{\bar{n}} | {\hat{r}}_{0} ({\tilde{Y}}_{j}) - {\hat{r}}_{1} ({\tilde{Y}}_{j}) |^{β}$ , where

{\hat{r}}_{s} (y) = \frac{\sum_{i = 1}^{\bar{n}} g ({\tilde{X}}_{i}) 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Y}}_{i} - y)}{\sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Y}}_{i} - y)}, s \in {0, 1},

for some

β > 0

, and

K

is some kernel with bandwidth

h

. We denote

K_{h}

as in Section 4.2, for any univariate or multivariate kernel

K

(c)

Lack of disparate mistreatment: the empirical $γ$ disparate mistreatment rate is

{\hat{D M R}}_{γ} := \frac{\sum_{j = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) - {\tilde{Y}}_{j} |^{γ} 1 ({\tilde{S}}_{i} = 0) / \sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = 0)}{\sum_{j = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) - {\tilde{Y}}_{j} |^{γ} 1 ({\tilde{S}}_{i} = 1) / \sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = 1)}, γ > 0.

As for the conditional fairness, we can calculate the previous measures when

Z_{1} = z

, for every value of

z

(d)

Conditional demographic parity: the empirical conditional $α$ -disparate impact given $Z_{1} = z$ is ${\hat{D I}}_{α} (z) := {\hat{r}}_{c, D P} (z, 0) / {\hat{r}}_{c, D P} (z, 1)$ , where

{\hat{r}}_{c, D P} (z, s) := \frac{\sum_{i = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) |^{α} 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Z}}_{i, 1} - z)}{\sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Z}}_{i, 1} - z)}, s \in {0, 1}, α > 0.

(e)

Conditional equalized odds: the empirical mean conditional $L^{β}$ fairness given $Z_{1} = z$ is ${\hat{M L F}}_{β} (z) := 1 - \sum_{j = 1}^{\bar{n}} | {\hat{r}}_{0} (z, {\tilde{Y}}_{j}) - {\hat{r}}_{1} (z, {\tilde{Y}}_{j}) |^{β} / \bar{n}$ , where

{\hat{r}}_{s} (z, y) = \frac{\sum_{i = 1}^{\bar{n}} g ({\tilde{X}}_{i}) 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Y}}_{i} - y, {\tilde{Z}}_{i, 1} - z)}{\sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = s) K_{h} ({\tilde{Y}}_{i} - y, {\tilde{Z}}_{i, 1} - z)}, s \in {0, 1}, β > 0.

(f)

Conditional lack of disparate mistreatment: the empirical conditional $γ$ disparate mistreatment rate given $Z_{1} = z$ is ${\hat{D M R}}_{γ} (z) := {\hat{r}}_{c, L D M} (z, 0) / {\hat{r}}_{c, L D M} (z, 1)$ , where

{\hat{r}}_{c, L D M} (z, s) := \frac{\sum_{j = 1}^{\bar{n}} | g ({\tilde{X}}_{i}) - {\tilde{Y}}_{j} |^{γ} 1 ({\tilde{S}}_{i} = 0) K_{h} ({\tilde{Z}}_{i, 1} - z)}{\sum_{i = 1}^{\bar{n}} 1 ({\tilde{S}}_{i} = 0) K_{h} ({\tilde{Z}}_{i} - z) K_{h} ({\tilde{Z}}_{i, 1} - z)}, γ > 0.

To obtain a global idea of conditional fairness, we can calculate an “average” of these measures to estimate

E [D I_{α} (Z_{1})]

E [M L F_{β} (Z_{1})]

and

E [D M R_{γ} (Z_{1})]

as follows:

{\hat{D I}}_{α}^{t o t} := \frac{1}{\bar{n}} \sum_{i = 1}^{\bar{n}} {\hat{D I}}_{α} ({\tilde{Z}}_{i}), {\hat{M L F}}_{β}^{t o t} := \frac{1}{\bar{n}} \sum_{i = 1}^{\bar{n}} {\hat{M L F}}_{β} ({\tilde{Z}}_{i}), {\hat{D M R}}_{γ}^{t o t} := \frac{1}{\bar{n}} \sum_{i = 1}^{\bar{n}} {\hat{D M R}}_{γ} ({\tilde{Z}}_{i}) .

If the conditional variable

Z_{1}

is discrete, just replace

K_{h} ({\tilde{Z}}_{i} - z)

with

1 ({\tilde{Z}}_{i} = z)

in the definitions above. Hereafter, we set

α = β = γ = 1

. Moreover, the powers

a

and

b

that appeared in the optimisation programmes of Section 3 will be equal to

2

Remark 5.1

The choice of the bandwidth $h$ is of practical importance when dealing with continuous outcomes or explanatory variables. This is the case when $Y$ is continuous (EO) or when $Z_{1}$ has some continuous components (conditional fairness). A solution would be to rely on one of the bandwidth selectors that have been proposed in the kernel regression literature (cross-validation, plug-in, etc). See the survey of Vieu (1993), e.g. Alternatively, we could optimize our penalised criteria w.r.t. $(ω, h)$ instead of $ω$ . This can be justified by the uniform in bandwidth convergence of kernel-type regressors (Einmahl & Mason, 2005, Corollary 2). The bandwidth $h$ has just to belong to a compact interval $[a_{n}, b_{n}]$ , where $b_{n} \to 0$ and the rate $a_{n}$ depends on the regularity of the model.

5.2. Real Data Classification Experiments

We use two commonly-used real datasets for fairness studies: Adult dataset⁸ and COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) dataset.⁹ The Adult database provides information about gender, age, education level, etc., of some individuals. The task is to predict whether a person can gain more than 50k USD per year, using gender as the sensitive variable. The COMPAS recidivism dataset includes information about criminal history information, gaol and prison time, demographic features, etc. The task is to predict if a prisoner is likely to have recidivist behaviour, with race as the protected attribute. Here, we distinguish only the white community from the non-white community, without further digging into the precise race classification. A summary of the two latter databases is given in Table 1. After some basic preprocessing of these two datasets (dealing with the missing data, encoding the categorical variables, etc.), we split each dataset into three parts: training (30%), calibration (50%) and test (20%) datasets. We compare the performance of the bagging predictor (without any discretization procedure and without imposing the positivity of the weights) with other state-of-the-art post-processing fair learning methods, especially the methods of Agarwal et al. (2018) and that of Hardt et al. (2016). The latter methods yields a way of improving the fairness of some basis predictors. We have used the existing code provided by these authors. For comparison purpose, the training datasets and the test datasets are the same among different post-processing methods.

To illustrate the method, let us specify the criterion to be minimised in the DP case and for the COMPAS dataset. Here, we consider 5 basis predictors.¹⁰ Any vector $X_{i}$ of covariates yields the individual characteristics of a defendant and every basis predictor provides a forecasted value of $i$ ’s probability of recidivism knowing $X_{i}$ , say $r_{k} (X_{i})$ , $k \in {1, \dots, 5}$ . If $i$ has been really a repeat offender, then $Y_{i} = 1$ ; otherwise, $Y_{i} = 0$ . For a given vector of weights $ω \in R^{5}$ , we seek a new predictor ${\hat{Y}}_{ω}$ that is a linear combination of the basis ones, i.e. such that

{\hat{Y}}_{ω} (X_{i}) = ω_{1} r_{1} (X_{i}) + \dots + ω_{5} r_{5} (X_{i}),

for any

i \in {1, \dots, \tilde{n}}

. Denote

N_{\tilde{n}} (S = 1)

N_{\tilde{n}} (S = 0)

N_{\tilde{n}} (S = 1, {\hat{Y}}_{ω} = 1)

and

N_{\tilde{n}} (S = 0, {\hat{Y}}_{ω} = 1)

the number of individuals in the test sample of size

\tilde{n}

for which

S = 1

(black people),

S = 0

(not black),

(S, {\hat{Y}}_{ω}) = (1, 1)

(black and expected to be recidivist) and

(S, {\hat{Y}}_{ω}) = (0, 1)

(not black and expected to be recidivist), respectively. Then our optimal vector of weights

ω \in R^{5}

is the minimiser of the map

ω \mapsto {(\frac{N_{\tilde{n}} (S = 0, {\hat{Y}}_{ω} = 1)}{N_{\tilde{n}} (S = 0)} - \frac{N_{\tilde{n}} (S = 1, {\hat{Y}}_{ω} = 1)}{N_{\tilde{n}} (S = 1)})}^{2} + \frac{λ}{\tilde{n}} \sum_{i = 1}^{\tilde{n}} ({\hat{Y}}_{ω} - Y_{i})^{2},

imposing

ω_{1} + \dots + ω_{5} = 1

In Figures 2 and 3, we use a 2-dimension map to compare the performances of all these different predictors. One dimension evaluates fairness (with perfect fairness at zero), and the other dimension evaluates precision with MSE. The closer the performance of a model is to $(0, 0)$ , the better the predictor. We show also the performance of our bagging estimator for a wide range of the tuning parameter $λ$ values. This is indicated by different colours (see the colour bar shown on the right-hand side). Depending on the value of $λ$ , we observe an expected trade-off between prediction accuracy and fairness. The closer a predictor is to the south-west corner of a figure, the better the latter trade-off it proposes. Generally speaking, the considered basis predictors perform well in terms of accuracy (small MSE), but behave poorly in terms of fairness. Moreover, the ”fair estimators” obtained by Agarwal et al. (2018)’s method and Hardt et al. (2016)’s method are more fair than the latter ones, at the price of a reasonable loss in terms of accuracy. No clear-cut hierachy appears between the Agarwal et al. (2018)’s predictors and Hardt et al. (2016)’s ones. As our graphs suggest, our predictors “by averaging” most often perform better than all the latter families of predictor, for a wide range of hyperparameter $λ$ values: most of the alternative predictors lie to the north-east of the piecewise linear curve that links the performances of our predictors by averaging, which justifies their relevance.

Figure 2.

Comparison of performances between our bagging predictor and other post-processing methods (COMPAS dataset). (a) Demographic Parity performance of the bagging predictor on the COMPAS dataset, (b) Equalized Odds performance of the bagging predictor on the COMPAS dataset, (c) Lack of Disparate Mistreatment performance of the bagging predictor on the COMPAS dataset.

Figure 3.

Comparison of performances between our bagging predictor and other post-processing methods (Adult dataset). (a) Demographic Parity performance of the bagging predictor on the Adult dataset, (b) Lack of Disparate Mistreatment performance of the bagging predictor on the Adult dataset.

5.3. Real Data Regression Experiments

In the case of continuous explained variables $Y$ , we have to manage regression models instead of classification ones. We led the same experiment as in the previous section using two real datasets: the Crime dataset¹¹ and the Law dataset.¹² The former dataset contains law enforcement, crime and socio-economic data of communities in the US. The task is to predict violent crimes number per 100 k population. Here, the protected variable indicates if the habitants of a given community are mostly white. The Law dataset contains some admission data of two thirds of the public law schools in the US. The task is here to predict a student’s GPA with race as the sensitive feature. A summary of these two databases is provided in Table 2.

Table 2.
Summary of the “Crime” and “Law” Datasets.

Name Number of Observations Sensitive Feature Target Variable

Crime 1944 White_community Violent crimes per population

Law 24456 Is_white GPA

Name	Number of Observations	Sensitive Feature	Target Variable
Crime	1944	White_community	Violent crimes per population
Law	24456	Is_white	GPA

After some basis preprocessing of these two datasets as in Section 5.2, we similarly split each dataset into three parts (training, calibration and test subsets). In the case of regression, there is no discretization issue, and optimisation could be done easily. Thus, we have tested the bagging method with different fairness definitions, as discussed before.

With the same notation as in Section 5.2 and to illustrate, let us specify the criterion to be minimised in the EO case and for the Law dataset. We still invoke the 5 basis predictors as for classification purpose. Every basis predictor provides a forecasted value of $i$ ’s GPA knowing $X_{i}$ , say $r_{k} (X_{i})$ , $k \in {1, \dots, 5}$ . The true GPA of $i$ is a real number $Y_{i}$ , as our new predictor ${\hat{Y}}_{ω}$ for any vector of weights $ω \in R^{5}$ . The sensitive feature $S_{i}$ is equal to one if and only if $i$ is white. We choose a Gaussian kernel and its bandwidth $h$ is the empirical standard deviation of $(Y_{i})_{i = 1}^{\tilde{n}}$ divided by ${\tilde{n}}^{1 / 5}$ (the usual “rule of thumb”). Define

q (s, y, ω) = \frac{\sum_{i = 1 s.t. S_{i} = s}^{\tilde{n}} {\hat{Y}}_{ω} (X_{i}) \exp (- (Y_{i} - y)^{2} / 2 h^{2})}{\sum_{i = 1 s.t. S_{i} = s}^{\tilde{n}} \exp (- (Y_{i} - y)^{2} / 2 h^{2})},

for

s \in {0, 1}

. Then our optimal vector of weights

ω \in R^{5}

is the minimiser of

ω \mapsto \sum_{i = 1}^{\tilde{n}} (q (0, Y_{i}, ω) - q (1, Y_{i}, ω))^{2} + λ \sum_{i = 1}^{\tilde{n}} ({\hat{Y}}_{ω} (X_{i}) - Y_{i})^{2}, with ω_{1} + \dots + ω_{5} = 1.

We still want to compare the performance of the bagging predictor with other post-processing fair learning methods. As fairness in regression models is much less discussed than in classification ones, and since the relevant literature rarely provides codes, we decided to compare our bagging predictor only with the Agarwal et al. (2019)’s predictor. For this comparison, the training and test datasets are the same among the different methods. In Figures 4 and 5, we use the same 2-dimension maps as in Section 5.2. As our figures suggest, the bagging regressor ameliorates the fairness of the basis estimators for a wide range of

λ

values. Unlike the case of classification, the performances of Agarwal et al. (2019)’s regressors are of the same order as that obtained with our method by averaging. Indeed, Agarwal et al. (2019)’s predictors are often close to the piecewise linear curve which describes the performance of our own predictors for the Crime dataset. Things are less clear-cut in the case of the Law dataset. Nonetheless, the flexibility of adjusting our performances for a wide range of

λ

values is an advantage of the bagging method compared to our competitors. The R code for the empirical sections is available at https://gitlab.com/lxwgale/fairbagging

Figure 4.

Comparison of performances between our bagging predictor and other post-processing methods (Crime dataset). (a) Demographic Parity performance of the bagging regressor on the Crime dataset, (b) Equalized Odds performance of the bagging regressor on the Crime dataset, (c) Lack of Disparate Mistreatment performance of the bagging predictor on the Crime dataset.

Figure 5.

Comparison of performances between our bagging predictor and other post-processing methods (Law dataset). (a) Demographic Parity performance of the bagging regressor on the Law dataset, (b) Lack of Disparate Mistreatment performance of the bagging predictor on the Law dataset.

6. Conclusion

We have studied a new simple method to improve fairness from a set of several “basis” unfair classifiers/regressors, by linearly combining their forecasts. Our procedure may be considered as a particular case of “model averaging” or “bagging”, to be inline with the terminology in machine learning. Such ideas have been applied to three standard fairness concepts - Demographic Parity, Equalized Odds and Lack of Disparate Mistreatment -, in addition to some extensions towards “conditional fairness” given the values of some non sensitive features. In practical terms, there is a trade-off between accuracy and fairness, as expected. Even with a small number of predictors, an almost perfect fairness seems to be reachable most often, but at the price of a significant loss of accuracy. Clearly, the more basis predictors we have, the more flexible the model will be. And the easier it will be to achieve near perfect fairness at a reasonable price in terms of accuracy. A simulation study has shown the potential of our method and its sensitivity to our tuning parameter. Our method performs more than honourably compared to some competing methods, as observed on several real databases. Nonetheless, more empirical work is surely necessary to evaluate to what extent our findings and intuitions are robust in practise. Moreover, finding an “optimal” data-driven tuning parameter is another avenue for further studies. This choice would surely depend on the individual preferences of users (and society as a whole) between fairness and forecasting performances. Under this perspective, a true behavioural model seems to be necessary.

We have improved a set of basic predictors by simple linear combinations. One advantage of this approach is its simplicity, but it can also be seen as a limitation. Thus, it could be desirable to extend our framework by looking for non-linear functions of the first stage predictors. In principle, this would be theoretically possible, at the cost of more complex and relatively arbitrary parameterizations, which are open to question. What is more, we would lose the closed-form solutions we had found. Nevertheless, we keep this idea for future work.

Footnotes

ORCID iD

Jean-David Fermanian

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: J.-D. Fermanian was supported by the labex Ecodec (reference project ANR-11-LABEX-0047).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Notes

A simulation Experiment

Let us evaluate the numerical performances of our “fair bagging” predictors on a simulated experiment. To this aim, we generate a synthetic dataset where the explained variable $Y$ and the sensitive feature $S$ are binary, and we rely on a bivariate continuous random vector $Z$ . We use the same Data Generating Process as in Donini et al. (2018). Given the value of the couple $(Y, S)$ , they propose to generate $Z$ from different bivariate Gaussian vectors. To be specific, given $(Y, S) = (1, 1)$ , the law of $Z$ is drawn $N ((- 1, - 1), 0.8 I_{2})$ . This will be the case for $5 / 16$ of the observations. Given $(Y, S) = (1, 0)$ , the law of $Z$ is drawing along a $N ((1, 1), 0.8 I_{2})$ . This situation will occur for $1 / 16$ of the sampled vectors. Given $(Y, S) = (0, 1)$ (resp. $(Y, S) = (0, 0)$ ), $Z \sim N ((0.5, - 0.5), 0.5 I_{2})$ (resp. $Z \sim N ((0.5, 0.5), 0.5 I_{2})$ ), for $5 / 16$ of the sampled vectors. Therefore, the realizations of the bivariate vector $Z$ clearly depend on $S$ . Thus, it is very likely that some performing basis classifiers that only use the information $Z$ (but not $S$ ) as inputs to predict $Y$ will be unfair. See Friedler et al. (2019) for alternatives simulation schemes.

For every experiment, as a preliminary stage, we evaluate $M = 3$ basis classifiers chosen as follows: a logistic model, a regression tree (as given by the rpart procedure in R) and a simple single hidden-layer neural network with two neurons. These three models are independently fitted on a particular sample of size $1000$ . Then, we independently draw a calibration i.i.d. sample $S_{n} := (X_{i}, Y_{i})_{i = 1, \dots, n}$ , $n = 5000$ . Our predictors by model averaging are estimated on $S_{n}$ , as discussed in Section 3 in the main document. Finally, we evaluate their performances in terms of predictive power (accuracy) and fairness out-of-sample, by calculating the empirical counterparts of the measures given in Section 4.1 of the core paper from a sample $T_{\bar{n}}$ , $\bar{n} = 5000$ , that is independent of $S_{n}$ . Predicted values ${\hat{Y}}_{i}$ are then available for measuring the in-sample (resp. out-of-sample) accuracy of our predictor on $S_{n}$ (resp. $T_{\bar{n}}$ ). Here, we have selected the empirical mean squared error or MSE as our measure of accuracy: $d = d_{2}$ , with the notations of Section 3.2 in the main document.

To evaluate the performances of the new bagging classifiers, we are free to choose continuous and/or discrete predictors, during the calibration stage and/or the testing stages, as explained in Section 3.3 of the main paper. To simplify, we consider only two cases:

the “no-discretization” case, where our predictor by model averaging is a real number, directly calculated as a linear combination of the outputs of the basis classifiers, for calibration and testing purposes ( $k = l = 0$ with the notations of Section 3.3), using the numerical smoothing trick explained at the end of Section 3.3;

the discretised case, where the bagging predictor is always binary, during the calibration and the testing stages ( $k = l = 1$ ).

The whole procedure is replayed

250

times. We calculate the median of the empirical measures defined in Section 4.1, for the three previous basis classifiers and for the new classifiers after model averaging. These figures for the basis classifiers are summarized in Table 3. Note that they are clearly “unfair”, as expected. Table 3.

Accuracy and Fairness of the Three Basis Classifiers, that have been Combined by Model Averaging.

Fairness	Logistic		Tree		Neural Networks
	Accur.	Fairness	Accur.	Fairness	Accur.	Fairness
DP (no discr.)	0.0718	0.331	0.077	0.343	0.138	0.362
	(0.0016)	(0.0199)	(0.0018)	(0.0231)	(0.0372)	(0.1325)
DP (discr.)	0.096	0.202	0.104	0.173	0.146	0.201
	(0.0037)	(0.0180)	(0.0068)	(0.0923)	(0.0374)	(0.1022)
EO (no discr.)	0.070	0.905	0.079	0.903	0.130	0.861
	(0.0022)	(0.0069)	(0.0041)	(0.0097)	(0.0107)	(0.0736)
EO (discr.)	0.097	0.868	0.104	0.866	0.147	0.844
	(0.0040)	(0.0149)	(0.0058)	(0.05967)	(0.0108)	(0.0799)
LDM (no discr.)	0.073	2.438	0.079	1.826	0.143	2.236
	(0.0014)	(0.1541)	(0.0039)	(0.2525)	(0.0180)	(0.2584)
LDM (discr.)	0.096	2.506	0.104	1.995	0.149	2.149
	(0.0041)	(0.2134)	(0.0053)	(0.2996)	(0.0267)	(0.2806)

For every type of fainess concept, the medians (first line) and the standard deviations (second line, into brackets) of the corresponding performances measures (as defined in section 4.1) are calculated on 250 independent drawns.

See Figures 6 to 8 for the corresponding performances of our bagging classifiers. In the latter case, the trade-off between accuracy and fairness strongly depends on the value of the tuning parameter

λ

. For small values of

λ

, we obtain average classifiers that are close to fairness, even if the basis classifiers were far from fairness. Nonetheless, their predictive power may be poor, at least poorer than those of the basis classifers. When

λ

is large, this is the opposite: our bagging classifiers exhibit a better accuracy, but at the price of a poor fairness. Figure 6.

Measure of accuracy (Mean Square Error; dashed line) and measures of Demographic Parity (disparate impact $D I_{1}$ ; solid line) for bagging classifiers, as a function of $\log_{10} (λ)$ . The optimisation programme is (3.11) with $(k, l) = (0, 0)$ (Top: no discretization of our classifier) or $(k, l) = (1, 1)$ (Bottom: discretization of our classifier). Perfect fairness corresponds to $D I_{1} = 1$ . Perfect accuracy corresponds to $M S E = 0$ .

Figure 7.

Measure of accuracy (Mean Square Error; dashed line) and measures of Equalized Odds (Mean $L^{2}$ fairness $M L F_{2}$ ; solid line) for classifiers by model averaging, as a function of $\log_{10} (λ)$ . The optimisation programme is (3.12) with $a = 2$ , $(k, l) = (0, 0)$ (Top: no discretization of our classifier by model averaging) or $(k, l) = (1, 1)$ (Bottom: discretization of our classifier by model averaging). Perfect fairness corresponds to $M L F_{2} = 1$ . Perfect accuracy corresponds to $M S E = 0$ .

Figure 8.

Measure of accuracy (Mean Square Error; dashed line) and measures of Lack of Disparate Mistreatment (disparate mistreatment rate $D M R_{2}$ ; solid line) for classifiers by model averaging, as a function of $\log_{10} (λ)$ . The optimisation programme is (3.13) with $a = b = 2$ , $(k, l) = (0, 0)$ (Top: no discretization of our classifier by model averaging) or $(k, l) = (1, 1)$ (Bottom: discretization of our classifier by model averaging). Perfect fairness corresponds to $D M R_{2} = 1$ . Perfect accuracy corresponds to $M S E = 0$ .

Generally speaking, it is difficult to guess an “optimal value” of $λ$ , that would simultaneously insure “reasonable” levels of fairness and accuracy. In the case of DP (Figure 6), it would make sense to select this tuning parameter so that $\log_{10} (λ) ≃ - 1.5$ (no discretization case). With discretised classifiers, DP fairness is always rather low when accuracy performs well, and no tuning parameter would realize a nice trade-off. For EO and Figure 7, we could promote $\log_{10} (λ) ≃ - 1$ without discretization and $\log_{10} (λ) ≃ 0$ with discretization. Concerning LDM and Figure 8, $\log_{10} (λ) ≃ - 1$ seems to be a good choice without discretization. Note that we obtain estimators close to perfect fairness when $λ$ tends to zero, whatever the definition of fairness, with or without discretization of our predictors. The single exception is related to LDM with discretization, that remains unfair whatever the value of $λ$ : $D M R_{2}$ is staying around $2$ whatever the value of the tuning parameter, namely far away the target value of one. This means that our methodology may fail under some circumstances, when the convex hull of the considered basis classifiers is not sufficiently rich to obtain perfect fairness. Finally, we have observed that significantly better results are obtained without discretizing our forecasts. Intuitively, this makes sense because optimising a smoother criterion should indeed generate more stable performances out-of-sample.

Remark A.1 In practical terms, a naive approach would be to select our tuning parameter to reach an a priori given target in terms of fairness, for some given definition and some measure of the latter concept. Nonetheless, a more formal approach could be proposed. Any user (a company, typically) has to evaluate two types of risks and the associated costs: the risk of unfairness (potential lawsuits and fines, regulatory issues, reputational damages, etc) on one side, and the risk of poor forecasts on the other side (loss of efficiency). For any value of $λ$ , one gets an optimal vector $ω (λ)$ and a corresponding bagging predictor. If this predictor is implemented, a user will suffer an expected loss $F (λ)$ due to a lack of fairness, in addition to an expected loss $A (λ)$ due to a lack of accurracy (prediction mistakes). Then, a theoretically optimal value of $λ$ could minimize a weighted average $α F (λ) + (1 - α) A (λ)$ for some given $α \in (0, 1)$ , whose value depends on the relative preferences for accuracy and fairness. The spirit of such a method is be close to the tuning parameter selector proposed by Centorrino et al. (2022), equation (4.22).

Solution of Programme (3.8)

In the case of lack of disparate mistreatment and setting $a = b = 2$ , the optimisation programme (3.8) may be written (B.1)

{\hat{ω}}^{*} := \arg min_{ω} {(ω^{'} Σ ω)^{2} + λ ω^{'} \bar{Σ} ω},

where

Σ := {\bar{Σ}}_{0} - {\bar{Σ}}_{1}

, and under the constraint

ω^{'} e = 1

. The associated lagrangian is

L (ω, μ) := (ω^{'} Σ ω)^{2} + λ ω^{'} \bar{Σ} ω + μ (1 - ω^{'} e),

for some scalar

μ

. First order conditions yield

4 (ω^{'} Σ ω) Σ ω + 2 λ \bar{Σ} ω = μ e .

Thus, the optimal value of

ω

is proportional to

(2 (ω^{'} Σ ω) Σ + \bar{Σ})^{- 1} e

. Since it has to satisfy

ω^{'} e = 1

, we get

{\hat{ω}}^{*} = \frac{(2 (ω^{'} Σ ω) Σ + λ \bar{Σ})^{- 1} e}{e^{'} (2 (ω^{'} Σ ω) Σ + λ \bar{Σ})^{- 1} e},

yielding the announced result (equation (3.9) in the core document).

References

Abebe

S. A.

Lucchese

Orlando

(2022). EIFFFEL: Enforcing fairness in forests by flipping leaves. In Proceedings of the 37th ACM/SIGAPP symposium on applied computing, (pp. 429–436).

Agarwal

Beygelzimer

Dudík

Langford

Wallach

(2018). A reductions approach to fair classification. In International conference on machine learning, (pp. 60–69). PMLR.

Agarwal

Dudík

Z. S.

(2019). Fair regression: Quantitative definitions and reduction-based algorithms. In International conference on machine learning, (pp. 120–129). PMLR.

Barocas

Hardt

Narayanan

(2023). Fairness in machine learning, limitations and opportunities. MIT press.

Berk

Heidari

Jabbari

Kearns

Roth

(2018). Fairness in criminal justice risk assessments: The state of the art. Sociological Methods and Research. On-line publication, https://doi.org/10.1177/0049124118782533

Bhaskaruni

Lan

(2019). Improving prediction fairness via model ensemble. In 2019 IEEE 31st International conference on tools with artificial intelligence (ICTAI), (pp. 1810–1814). IEEE.

Biau

(2012). Analysis of a random forests model. Journal of Machine Learning Research, 13, 1063–109.

Biau

Scornet

(2015). A random forest guided tour. Test, 25, 197–227.

Breiman

(1996). Bagging predictors. Machine Learning, 140, 26–123.

10.

Breiman

(2001). Random forest. Machine Learning, 45, 5–32.

11.

Buolamwini

Gebru

(2018). Gender shades: Intersectional accuracy disparities in commercial gender classification. In Proceedings of 2018 conference on fairness, accountability and transparency, FAT 2018, New York, USA (pp. 81, 77–91).

12.

Calders

Kamiran

Pechenizkiy

(2009). Building classifiers with independency constraints. 2009 IEEE International Conference on Data Mining Workshops, 13–18.

13.

Calders

Verwer

(2010). Three naive Bayes approaches for discrimination-free classification. Data Mining and Knowledge Discovery, 21, 277–292.

14.

Calmon

Wei

Vinzamuri

Ramamurthy

K. N.

Varshney

K. R.

(2017). Optimized pre-processing for discrimination prevention. Proc. 31st International Conference on Neural Information Processing Systems, 3995–4004.

15.

Centorrino

Florens

J. P.

Loubes

J.-M.

(2022). Fairness in Machine Learning and Econometrics. In Econometrics with machine learning, (pp. 217–250).

16.

Chouldechova

Benavides-Prado

Fialko

Vaithianathan

(2018). A case study of algorithm-assisted decision making in child maltreatment hotline screening decisions. ACM Conference on Fairness, Accountability, and Transparency (FAT2018), 134–148.

17.

Chzhen

Denis

Hebiri

Oneto

Pontil

(2020). Fair regression via plug-in estimator and recalibration with statistical guarantees. Advances in Neural Information Processing Systems, 33, 19137–19148.

18.

Cleary

T. A.

(1966). Test bias: Validity of the scholastic aptitude test for negro and white students in integrated colleges. ETS Research Bulletin Series, 2, i–23.

19.

Corbett-Davies

Pierson

Feller

Goel

Huq

(2017). Algorithmic decision making and the cost of fairness. In Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining, (pp. 797–806).

20.

Darlington

R. B.

(1971). Another look at cultural fairness. Journal of Educational Measurement, 8, 71–82.

21.

del Barrio

Gordaliza

Loubes

J.-M.

(2020). Review of mathematical frameworks for fairness in machine learning. arXiv preprint arXiv:2005.13755.

22.

Di Carlo

Nezami

Anahideh

Asudeh

(2023). FairPilot: An explorative system for hyperparameter tuning through the lens of fairness. arXiv preprint arXiv:2304.04679.

23.

Donini

Oneto

Ben-David

Shawe-Taylor

J. S.

Pontil

(2018). Empirical risk minimization under fairness constraints. Advances in Neural Information Processing Systems, 2791–2801.

24.

Dwork

Hardt

Pitassi

Reingold

Zemel

(2012). Fairness Through Awareness. In Proc. 3rd innovations of theoretical computer science, (pp. 214–226).

25.

Einhorn

H. J.

Bass

A. R.

(1971). Methodological considerations relevant to discrimination in employment testing. Psychological Bulletin, 75, 261.

26.

Einmahl

Mason

D. M.

(2005). Uniform in bandwidth consistency of kernel-type function estimators. The Annals of Statistics, 33, 1380–1403.

27.

Fantin

(2020). A distributed fair random forest (Doctoral dissertation).

28.

Feldman

Friedler

S. A.

Moeller

Scheidegger

Venkatasubramanian

(2015). Certifying and removing disparate impact. In Proc. 21th ACM SIGKDD international conference on knowledge discovery and data mining, (pp. 259–268).

29.

Friedler

S. A.

Scheidegger

Venkatasubramanian

Choudhary

Hamilton

E. P.

Roth

(2019). A comparative study of fairness-enhancing interventions in machine learning. In Proc. conference on fairness, accountability, and transparency, (pp. 329–338).

30.

Friedman

J. H.

Hall

(2007). On bagging and nonlinear estimation. Journal of statistical planning and inference, 1373(3), 669–683.

31.

Ghysels

Babii

Chen

Kumar

(2020). Binary choice with asymmetric loss in a Data-Rich environment: Theory and an application to racial justice (No. 15418). CEPR discussion papers.

32.

Grgić-Hlaa

Zafar

M. B.

Gummadi

K. P.

Weller

(2017). On fairness, diversity and randomness in algorithmic decision making. arXiv preprint arXiv:1706.10208.

33.

Guardieiro

Raimundo

M. M.

Poco

(2023). Enforcing fairness using ensemble of diverse Pareto-optimal models. Data Mining and Knowledge Discovery, 1–29.

34.

Hardt

Price

Srebro

(2016). Equality of opportunity in supervised learning. Advances in Neural Information Processing Systems, 3315–3323.

35.

Hastie

Tibshirani

Friedman

(2009). The elements of statistical learning: Data mining, inference, and prediction. Springer Science & Business Media, LLC.

36.

Liu

Wang

Lan

(2019). A distributed fair machine learning framework with private demographic data protection. In 2019 IEEE international conference on data mining (ICDM), (pp. 1102–1107).

37.

Hurlin

Perignon

Saurin

(2024). The fairness of credit scoring models. Management Science, Online.

38.

Iosifidis

Fetahu

Ntoutsi

(2019). FAE: A fairness-aware ensemble framework. In 2019 IEEE international conference on big data (Big Data), (pp. 1375–1380). IEEE.

39.

Kamiran

Calders

(2012). Data preprocessing techniques for classification without discrimination. Knowledge and Information Systems, 33, 1–33.

40.

Kamiran

Calders

Pechenizkiy

(2010). Discrimination aware decision tree learning. 2010 IEEE international conference on data mining, (pp. 869–874).

41.

Kamiran

Žliobait

Calders

(2013). Quantifying explainable discrimination and removing illegal discrimination in automated decision making. Knowledge and Information Systems, 35, 613–644.

42.

Kamishima

Akaho

Asoh

Sakuma

(2012). Fairness-aware classifier with prejudice remover regularizer. Joint European Conference on Machine Learning and Knowledge Discovery in Databases, 35–50.

43.

Kearns

Neel

Roth

Z. S.

(2018). Preventing fairness gerrymandering: Auditing and learning for subgroup fairness. arXiv preprint arXiv:1711.05144.

44.

Kenfack

P. J.

Khan

A. M.

Kazmi

S. A.

Hussain

Oracevic

Khattak

A. M.

(2021). Impact of model ensemble on the fairness of classifiers in machine learning. In 2021 International conference on applied artificial intelligence (ICAPAI), (pp. 1–6). IEEE.

45.

Kim

Moon

Ahn

(2011). A weight-adjusted voting algorithm for ensembles of classifiers. Journal of the Korean Statistical Society, 40, 437–449.

46.

Kitagawa

Sakaguchi

Tetenov

(2021). Constrained classification and policy learning. arXiv preprint arXiv:2106.12886.

47.

Kleinberg

Mullainathan

Raghavan

(2017). Inherent trade-offs in the fair determination of risk scores. 2017 Innovations in Theoretical Computer Science Conference (ITCS), 1–43.

48.

Kotsiantis

Pintelas

(2004). Combining bagging and boosting. International Journal of Computational Intelligence, 1, 324–333.

49.

Kusner

M. J.

Loftus

Russell

Silva

(2017). Counterfactual fairness. Advances in Neural Information Processing Systems (NeurIPS 2017), 4066–4076.

50.

Lambrecht

Tucker

(2019). Algorithmic bias? An empirical study of apparent gender based discrimination in the display of stem career Ads. Management Science, 65, 2966–2981.

51.

Cui

Chen

(2017). Extracting optimal actionable plans from additive tree models. Frontiers of Computational Science, 11, 1–15.

52.

Lundberg

S. M.

Lee

S. I.

(2017). A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, 4765–4774.

53.

Luong

B. T.

Ruggieri

Turini

(2011). K-NN as an implementation of situation testing for discrimination discovery and prevention. In Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, (pp. 502–510).

54.

Madras

Creager

Pitassi

Zemel

(2018). Learning adversarially fair and transferable representations. In International conference on machine learnin, (pp. 3384–3393). PMLR.

55.

Makhlouf

Zhioua

Palamidessi

(2021). Machine learning fairness notions: Bridging the gap with real-world applications. Information Processing & Management, 58(5), 102642.

56.

Maudes

Rodriguez

J. J.

Garcia-Osorio

Garcia-Pedrajas

(2012). Random feature weights for decision tree ensemble construction. Information Fusion, 13, 20–30.

57.

Nanfack

Delchevalerie

Frénay

(2021). Boundary-based fairness constraints in decision trees and random forests. In ESANN 2021 Proceedings 29th European symposium on artificial neural networks, computational intelligence and machine learning. (pp. 375–380).

58.

Newey

W. K.

McFadden

(1994). Large sample estimation and hypothesis testing. In Handbook of econometrics 4, (pp. 2111–2245).

59.

Noble

S. U.

(2018). Algorithms of oppression: How search engines reinforce racism. (NYU Press).

60.

O’Neil

(2016). Weapons of math destruction: How big data increases inequality and threatens democraty. (Broadway Books).

61.

Pedreschi

Ruggieri

Turini

(2009). Integrating induction and deduction for finding evidence of discrimination. Proc. of International Conference on Artificial Intelligence and Law (ICAIL ’09), 157–166.

62.

Raff

Sylvester

Mills

(2018). Fair forests: Regularized tree induction to minimize model bias. In Proceedings of the 2018 AAAI/ACM conference on AI, ethics, and society, (pp. 243–250).

63.

Rambachan

Kleinberg

Mullainathan

Ludwig

(2020). An economic approach to regulating algorithms. Technical report, National Bureau of Economic Research.

64.

Ritov

Y. A.

Sun

Zhao

(2017). On conditional parity as a notion of non-discrimination in machine learning. arXiv preprint arXiv:1706.08519.

65.

Verma

Rubin

(2018). Fairness definitions explained. In Proceedings of the international workshop on software fairness, (pp. 1–7).

66.

Vieu

(1993). Bandwidth selection for kernel regression: A survey. Computer Intensive Methods in Statistics (Physica, Heidelberg), 134–149.

67.

Wager

Athey

(2018). Estimation and inference of heterogeneous treatment effects using random forests. Journal of The American Statistical Association, 113, 1228–1242.

68.

Williamson

R. C.

Menon

A. K.

(2019). Fairness risk measures. In International conference on machine learning, (pp. 6786–6797). PMLR.

69.

Woodworth

B. E.

Gunasekar

Ohannessian

M. I.

Srebro

(2017). Learning Non-discriminatory predictors. Proc. 30th COLT, 1920–1953

70.

Cui

Kuang

Zhou

Shen

Cui

(2020). Algorithmic decision making with conditional fairness. In Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, (pp. 2125–2135).

71.

Zafar

M. B.

Valera

Gomez Rodriguez

Gummadi

K. P.

(2017). Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment. In Proc. 26th International conference on world wide web, international world wide web conferences steering committee, (pp. 1171–1180).

72.

Zemel

Swersky

Pitassi

Dwork

(2013). Learning fair representations. In Proc. international conference on machine learning, (pp. 325–333).

73.

Zhang

Bifet

Zhang

Weiss

J. C.

Nejdl

(2021). FARF: A fair and adaptive random forests classifier. In Advances in knowledge discovery and data mining: 25th Pacific-Asia conference, PAKDD 2021, Proceedings, Part II, (pp. 245–256).

74.

Zliobaite

(2015). On the relation between accuracy and fairness in binary classification. arXiv preprint arXiv:1505.05723.

Fair Learning by Model Averaging

Abstract

Keywords

1. Introduction

2. Analysis of the Literature

Table 1. Summary of the “Adult” and “COMPAS” Datasets. Name Nb of Observations Sensitive Feature Target Variable Adult 45222 gender income > 50k USD COMPAS 6172 race recidivism

4.1. The Idea of Model Averaging

4.2. Penalized Criteria

5. Empirics

5.1. Performance Measures

Table 2. Summary of the “Crime” and “Law” Datasets. Name Number of Observations Sensitive Feature Target Variable Crime 1944 White_community Violent crimes per population Law 24456 Is_white GPA

Footnotes

ORCID iD

Funding

Declaration of Conflicting Interests

Notes

A simulation Experiment

Solution of Programme (3.8)

References

Table 1.
Summary of the “Adult” and “COMPAS” Datasets.

Name Nb of Observations Sensitive Feature Target Variable

Adult 45222 gender income > 50k USD

COMPAS 6172 race recidivism

Table 2.
Summary of the “Crime” and “Law” Datasets.

Name Number of Observations Sensitive Feature Target Variable

Crime 1944 White_community Violent crimes per population

Law 24456 Is_white GPA