Decision-Focused Optimal Transport

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.