Optimal Transport under Group Fairness Constraints

TL;DR

This paper introduces new methods for incorporating group fairness constraints into optimal transport problems, including efficient algorithms and relaxations, with theoretical guarantees and empirical validation.

Contribution

It proposes a novel fairness constraint for OT, a modified Sinkhorn algorithm, and two relaxation strategies with theoretical analysis and practical effectiveness.

Findings

Efficient fair OT plans via modified Sinkhorn algorithm.

Finite-sample guarantees for penalized OT approach.

Bound on fairness deviation in bilevel optimization.

Abstract

Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalized OT problem, for which we derive novel finite-sample complexity guarantees. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound on the deviation of fairness when matching unseen data. Finally, we present empirical results illustrating the performance of our approaches and the trade-off between fairness and transport cost.