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; 1 

