Decision-Focused Optimal Transport

TL;DR

This paper introduces a decision-focused divergence metric tailored for stochastic linear optimization, which better captures cost discrepancies between distributions by considering the optimizer's sensitivity, and provides efficient computation and estimation guarantees.

Contribution

The paper proposes a novel decision-focused divergence metric based on optimal transport, specifically designed for stochastic linear optimization problems, with computational methods and sample complexity analysis.

Findings

The DF distance aligns better with cost differences than traditional metrics.

Efficient algorithms are developed for computing the DF distance.

Sample complexity guarantees show robustness in high dimensions.

Abstract

We propose a fundamental metric for measuring the distance between two distributions. This metric, referred to as the decision-focused (DF) divergence, is tailored to stochastic linear optimization problems in which the objective coefficients are random and may follow two distinct distributions. Traditional metrics such as KL divergence and Wasserstein distance are not well-suited for quantifying the resulting cost discrepancy, because changes in the coefficient distribution do not necessarily change the optimizer of the underlying linear program. Instead, the impact on the objective value depends on how the two distributions are coupled (aligned). Motivated by optimal transport, we introduce decision-focused distances under several settings, including the optimistic DF distance, the robust DF distance, and their entropy-regularized variants. We establish connections between the proposed DF distance and classical distributional metrics. For the calculation of the DF distance, we develop efficient computational methods. We further derive sample complexity guarantees for estimating these distances and show that the DF distance estimation avoids the curse of dimensionality that arises in Wasserstein distance estimation. The proposed DF distance provides a foundation for a broad range of applications. As an illustrative example, we study the interpolation between two distributions. Numerical studies, including a toy newsvendor problem and a real-world medical testing dataset, demonstrate the practical value of the proposed DF distance.