Optimal Transport under Group Fairness Constraints

TL;DR

The paper tackles group fairness in matching by casting the problem as optimal transport with planner-defined group constraints. It first introduces FairSinkhorn to compute exactly fair transport plans, then proposes two relaxations: fairness-penalized OT with a convex penalty and a cost-learning approach via bilevel optimization to induce fairness through the ground cost. It provides finite-sample guarantees for the penalized method and a generalization bound for the cost-learning approach on unseen data, plus empirical results demonstrating trade-offs between fairness and performance. The work advances principled, scalable fairness in OT with practical tools that can be reused for new samples and applications ranging from college admissions to resource allocation.

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 \texttt{FairSinkhorn}, 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 penalised OT problem, for which we derive novel finite-sample complexity guarantees. This result is of independent interest as it can be generalized to arbitrary convex penalties. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound guaranteeing that the learned cost yields fair matchings on unseen data. Finally, we present empirical results that illustrate the trade-offs between fairness and performance.

Figures (7)

• Figure 1: Summary of the fairness issues arising in optimal transport when sensitive features are highly correlated with features used for cost computations. On the left of the figure, we show an illustrative setting in which we wish to match two populations (circles and triangles). Group membership is highly correlated with the individual's location in the $(x,y)$ plane. This results in a block sparse transport plan, displayed on the right.
• Figure 2: Comparison of FairSinkhorn , penalized optimal transport, and Sinkhorn on the Gaussian data introduced in \ref{['ssec:simulated_data']}. FairSinkhorn achieves perfect fairness with a high transport cost while Sinkhorn achieves low transport cost with low fairness. The penalized OT interpolates between Sinkhorn and FairSinkhorn .
• Figure 3: Synthetic Gaussian data experiment. Left: 2D gaussians with binary sensitive attribute $S\in\{0,1\}$ and $W\in\{0,1\}$. Right: Utility/fairness tradeoffs. On the top, the utility metric is the KL divergence between the OT plan with and without the fairness constraint in function of the fairness regularization parameter. On the bottom, the difference between the transport cost with fair and unfair transport plan. The penalized OT correspond to our first relaxed approach while de Mahalanobis, MLP and ResNet ones correspond to the cost learning approach.
• Figure 4: Transport plan we get using different level of fairness parameter for each approach with the Gaussian data.
• Figure 5: Synthetic circular data experiment. Left: 2D gaussians with sensitive attribute $S\in\{0,1, 2\}$ and $W\in\{0,1, 2\}$. Right: Utility/fairness tradeoffs. On the top, the utility metric is the KL divergence between the OT plan with and without the fairness constraint in function of the fairness regularization parameter. On the bottom, the difference between the transport cost with fair and unfair transport plan. The penalized OT correspond to our first relaxed approach while de Mahalanobis, MLP and ResNet ones correspond to the cost learning approach.

Theorems & Definitions (22)

• Example 3.1
• Example 3.2
• Definition 3.3: $\mathbf{F}$-Fair coupling
• proof : Proof sketch for \ref{['th:sample_complexity_penalizedOT']}.
• Remark 4.1
• Example 4.1: Mahalanobis cost
• Example 4.2: Neural cost
• proof : Proof sketch for \ref{['thm:subgaussian_fluctuations']}
• proof
• Remark 1