FERMI: FAIR EMPIRICAL RISK MINIMIZATION VIA EXPONENTIAL R ÉNYI MUTUAL INFORMATION

Abstract

Several notions of fairness, such as demographic parity and equal opportunity, are defined based on statistical independence between a predicted target and a sensitive attribute. In machine learning applications, however, the data distribution is unknown to the learner and statistical independence is not verifiable. Hence, the learner could only resort to empirical evaluation of the degree of fairness violation. Many fairness violation notions are defined as a divergence/distance between the joint distribution of the target and sensitive attributes and the Kronecker product of their marginals, such as Rényi correlation, mutual information, L ∞ distance, to name a few. In this paper, we propose another notion of fairness violation, called Exponential Rényi Mutual Information (ERMI) between sensitive attributes and the predicted target. We show that ERMI is a strong fairness violation notion in the sense that it provides an upper bound guarantee on all of the aforementioned notions of fairness violation. We also propose the Fair Empirical Risk Minimization via ERMI regularization framework, called FERMI. Whereas existing in-processing fairness algorithms are deterministic, we provide a stochastic optimization method for solving FERMI that is amenable to large-scale problems. In addition, we provide a batch (deterministic) method to solve FERMI. Both of our proposed algorithms come with theoretical convergence guarantees. Our experiments show that FERMI achieves the most favorable tradeoffs between fairness violation and accuracy on test data across different problem setups, even when fairness violation is measured in notions other than ERMI.

1. INTRODUCTION

Ensuring that decisions made using machine learning algorithms are fair to different subgroups is of utmost importance. Without any mitigation strategy, machine learning algorithms may result in discrimination against certain subgroups based on sensitive attributes, such as gender or race, even if such discrimination is absent in the training data (Datta et al., 2015; Sweeney, 2013; Bolukbasi et al., 2016; Angwin et al., 2016; du Pin Calmon et al., 2017b; Feldman et al., 2015; Hardt et al., 2016; Fish et al., 2016; Woodworth et al., 2017; Zafar et al., 2017; Bechavod & Ligett, 2017; Kearns et al., 2018) . To remedy such discrimination issues, several notions for imposing algorithmic fairness have been proposed in the literature. A learning machine satisfies the demographic parity notion, if the predicted target is independent of the sensitive attributes (Dwork et al., 2012) . Promoting demographic parity can lead to poor performance, especially if the true outcome is not independent of the sensitive attributes. To remedy this, (Hardt et al., 2016) proposed equalized odds to ensure that the predicted target is conditionally independent of the sensitive attributes given the true label. A further relaxed version of this notion is equal opportunity which is satisfied if predicted target is conditionally independent of sensitive attributes given that the true label is in an advantaged class (Hardt et al., 2016) . Note that the inherent assumption in such conditional notions is that the true labels are unbiased. These notions suffer from a potential amplification of the inherent biases that may exist in the targets/labels in the training data (e.g., data collection bias). Tackling such bias is beyond the scope of this work. In practice, the learner cannot empirically verify independence of random variables, and hence cannot verify demographic parity, equalized odds, or equal opportunity. This has led the machine learning community to define several notions of fairness violation that quantify the degree of independence between random variables, e.g., demographic parity/equalized odds, L ∞ distance (Dwork et al., 2012; Hardt et al., 2016 ), mutual information (Kamishima et al., 2011; Rezaei et al., 2020; Steinberg et al., 2020; Zhang et al., 2018; Cho et al., 2020 ), Pearson correlation (Zafar et al., 2017) , false positive/negative rates (Bechavod & Ligett, 2017), Hilbert Schmidt independence criterion (HSIC) (Pérez-Suay et al., 2017) , and Rényi correlation (Baharlouei et al., 2020; Grari et al., 2020; 2019) , to name a few. In this paper, we define yet another notion of fairness violation, called exponential Rényi mutual information (ERMI). We show that ERMI is easy to compute empirically and prove that ERMI provides an upper bound on the existing notions of fairness violation such as demographic parity, equalized odds, and equal opportunity. Given a notion of fairness violation, it is still not straightforward to train an algorithm that satisfies a fairness violation constraint (Cotter et al., 2019) . The fairness-promoting machine learning algorithms can be categorized in three main classes: pre-processing, post-processing, and in-processing methods. Pre-processing algorithms (Feldman et al., 2015; Zemel et al., 2013; du Pin Calmon et al., 2017b) transform the biased data features to a new space in which the labels and sensitive attributes are statistically independent. This transform is oblivious to the training procedure. Post-processing approaches (Hardt et al., 2016; Pleiss et al., 2017) mitigate the discrimination of the classifier by altering the the final decision, e.g., by changing the thresholds on soft labels, or reassigning the labels to impose notions of fairness. In-processing approaches focus on the training procedure and impose the notions of fairness as constraints or regularization terms in the optimization procedure. Several regularization-based methods are proposed in the literature to impose measures of fairness to decision-trees (Kamiran et al., 2010; Raff et al., 2018; Aghaei et al., 2019) , support vector machines (Donini et al., 2018 ), neural networks (Grari et al., 2020) , or (logistic) regression models (Zafar et al., 2017; Berk et al., 2017; Taskesen et al., 2020; Chzhen & Schreuder, 2020; Baharlouei et al., 2020; Jiang et al., 2020; Grari et al., 2019) . To the best of our knowledge, existing in-processing methods are all deterministic, making them impractical for large-scale problems. Furthermore, most in-processing methods (with the exception of (Baharlouei et al., 2020) ) are designed for problems in which the sensitive attribute and/or the target is binary. In this paper, we introduce a new fair empirical risk minimization framework via ERMI regularization, and call it FERMI. We provide novel batch and stochastic gradient-based methods with guarantees for solving FERMI and demonstrate their effectiveness on multiple numerical experiments, which include a large-scale problem and problem with both non-binary sensitive attributes and targets. We show that FERMI can be used to achieve the most favorable tradeoffs between performance and fairness, even if fairness violation is measured in notions other than ERMI.

2. (Z, Z)-FAIRNESS: A GENERAL NOTION OF FAIRNESS

We consider a learner who trains a model to predict a target, Y , e.g., whether or not to extend a loan, supported on Y which can be discrete or continuous. The prediction is made using a set of features, X, e.g., financial history features, length of credit, and amount of debt. We also assume that there is a set of discrete sensitive attributes, S, e.g., race and sex, supported on S, associated with each sample. Further, let A ⊆ Y denote an advantaged outcome class, e.g., the outcome where a loan is extended. Next, we will present the main fairness notion considered in this paper, which generalizes several existing ones. Definition 1 ((Z, Z)-fairness). Given a random variable Z, let Z be a subset of values that Z can take. We say that a learning machine satisfies (Z, Z)-fairness if for every z ∈ Z, Y is conditionally independent of S given Z = z. More precisely, p Y ,S|Z ( y, s|z) = p Y |Z ( y|z)p S|Z (s|z) ∀ y ∈ Y, s ∈ S, z ∈ Z. (1) Notice that (Z, Z)-fairness recovers several important existing notions of fairness as special cases: 1. (Z, Z)-fairness recovers demographic parity (Dwork et al., 2012) if Z = 0 and Z = {0}. In this case, conditioning on Z has no effect, and hence (0, {0}) fairness is equivalent to the independence between Y and S, i.e., demographic parity (see Definition 7, Appendix A). 2. (Z, Z)-fairness recovers equalized odds (Hardt et al., 2016) if Z = Y and Z = Y. In the case, Z ∈ Z is trivially satisfied and could be dropped. Hence, conditioning on Z is equivalent to conditioning on Y, which recovers the equalized odds notion of fairness, i.e., conditional independence of Y and S given Y (see Definition 8, Appendix A). 3. (Z, Z)-fairness recovers equal opportunity (Hardt et al., 2016) if Z = Y and Z = A. This is also similar to the previous case with Y replaced with A (see Definition 9, Appendix A).

