决策导向最优传输
Decision-Focused Optimal Transport
AI总结：
本文提出决策导向最优传输，用于解决随机线性优化问题中的分布距离度量，通过引入DF发散度及其变体，解决了传统度量不足的问题，并展示了其在实际应用中的有效性。
摘要：
我们提出了一种用于衡量两个分布之间距离的基本度量。这种度量称为决策导向（DF）发散度，专门针对那些目标系数是随机且可能遵循两种不同分布的随机线性优化问题。传统的度量如KL发散度和瓦瑟斯坦距离并不适合量化由此产生的成本差异，因为系数分布的变化不一定改变底层线性规划的优化器。相反，对目标值的影响取决于两个分布如何耦合（对齐）。受最优传输启发，我们在几种设置下引入了决策导向距离，包括乐观DF距离、鲁棒DF距离及其熵正则化变体。我们建立了所提出的DF距离与经典分布度量之间的联系。对于计算DF距离，我们开发了高效的计算方法。我们进一步推导了估计这些距离的样本复杂性保证，并显示DF距离估计避免了瓦瑟斯坦距离估计中出现的维度灾难。所提出的DF距离为广泛的应用提供了基础。作为示例，我们研究了两个分布之间的插值。数值研究，包括一个玩具新供应商问题和一个真实世界的医疗测试数据集，展示了所提出DF距离的实用价值。
英文摘要：
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.
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