Optimal Pre-Processing to Achieve Fairness and Its Relationship with Total Variation Barycenter

TL;DR

This paper establishes a theoretical framework linking fairness, utility, and data pre-processing through total variation distances, and proposes an efficient linear programming approach to optimize fairness while controlling utility loss.

Contribution

It introduces a novel linear programming formulation for optimal pre-processing to enforce fairness based on total variation bounds, connecting fairness with barycenters of distributions.

Findings

Disparate impact is bounded by total variation distance between input distributions.

Utility degradation is bounded by total variation distance after pre-processing.

Optimal pre-processing can be efficiently computed via linear programming.

Abstract

We use disparate impact, i.e., the extent that the probability of observing an output depends on protected attributes such as race and gender, to measure fairness. We prove that disparate impact is upper bounded by the total variation distance between the distribution of the inputs given the protected attributes. We then use pre-processing, also known as data repair, to enforce fairness. We show that utility degradation, i.e., the extent that the success of a forecasting model changes by pre-processing the data, is upper bounded by the total variation distance between the distribution of the data before and after pre-processing. Hence, the problem of finding the optimal pre-processing regiment for enforcing fairness can be cast as minimizing total variations distance between the distribution of the data before and after pre-processing subject to a constraint on the total variation distance between the distribution of the inputs given protected attributes. This problem is a linear program that can be efficiently solved. We show that this problem is intimately related to finding the barycenter (i.e., center of mass) of two distributions when distances in the probability space are measured by total variation distance. We also investigate the effect of differential privacy on fairness using the proposed the total variation distances. We demonstrate the results using numerical experimentation with a practice dataset.