An Information Geometric Approach to Fairness With Equalized Odds Constraint

TL;DR

This paper introduces an information geometric framework for designing fair mechanisms that satisfy equalized odds without direct access to sensitive attributes, using mutual information approximations and quadratic programming.

Contribution

It develops a novel geometric approach to fairness that handles indirect sensitive attribute access and provides closed-form solutions and bounds for the mechanism design problem.

Findings

Quadratic programming solutions for fair mechanism design.

Bounds based on maximum singular value for approximate solutions.

Numerical comparisons showing effectiveness of the proposed methods.

Abstract

We study the statistical design of a fair mechanism that attains equalized odds, where an agent uses some useful data (database) $X$ to solve a task $T$. Since both $X$ and $T$ are correlated with some latent sensitive attribute $S$, the agent designs a representation $Y$ that satisfies an equalized odds, that is, such that $I(Y;S|T) =0$. In contrast to our previous work, we assume here that the agent has no direct access to $S$ and $T$; hence, the Markov chains $S - X - Y$ and $T - X - Y$ hold. Furthermore, we impose a geometric structure on the conditional distribution $P_{S|Y}$, allowing $Y$ and $S$ to have a small correlation, bounded by a threshold. When the threshold is small, concepts from information geometry allow us to approximate mutual information and reformulate the fair mechanism design problem as a quadratic program with closed-form solutions under certain constraints. For other cases, we derive simple, low-complexity lower bounds based on the maximum singular value and vector of a matrix. Finally, we compare our designs with the optimal solution in a numerical example.