协变量偏移下的情境优化：一种基于 Wasserstein 球交集的鲁棒方法
Contextual Optimization under Covariate Shift: A Robust Approach by Intersecting Wasserstein Balls
AI总结：
针对协变量偏移导致情境决策次优的问题，提出 IW-DRO 框架，通过非参数与参数估计器构建 Wasserstein 球交集模糊集，转化为凸规划求解，并证明了其优于单球模型的性能与有限样本集中性。
摘要：
在情境优化中，决策者利用情境信息（通常称为协变量）来更好地解决不确定性并做出明智决策。本文研究了协变量偏移下情境决策的挑战，协变量偏移是指协变量的分布在训练和测试环境之间存在差异的现象。这种偏移可能导致对远离训练数据的测试协变量的上游估计不准确，最终导致下游决策次优。为了应对这些挑战，我们提出了一种名为 Intersection Wasserstein-balls DRO (IW-DRO) 的新方法，该方法将多种估计方法集成到分布鲁棒优化 (DRO) 框架中。我们方法的核心是一个创新的模糊集，定义为两个 Wasserstein 球的交集，其中心使用适当的非参数和参数估计器构建。在计算方面，我们将 IW-DRO 问题重新表述为可处理的凸规划，并开发了一种针对大规模问题的近似算法以提高计算效率。在理论方面，我们证明了 IW-DRO 与单个 Wasserstein 球 DRO 模型相比具有更优的性能。我们进一步通过分析交集模糊集的覆盖范围以及两个估计器在 Wasserstein 距离下的测度集中性来建立性能保证。值得注意的是，我们推导出了协变量偏移下 Nadaraya-Watson 核估计器的有限样本集中结果。所提出的 IW-DRO 框架为在受协变量偏移影响的不确定环境中运作的决策者提供了实用价值。
英文摘要：
In contextual optimization, a decision-maker leverages contextual information, often referred to as covariates, to better resolve uncertainty and make informed decisions. In this paper, we examine the challenges of contextual decision-making under covariate shift, a phenomenon where the distribution of covariates differs between the training and test environments. Such shifts can lead to inaccurate upstream estimations for test covariates that lie far from the training data, ultimately resulting in suboptimal downstream decisions. To tackle these challenges, we propose a novel approach called Intersection Wasserstein-balls DRO (IW-DRO), which integrates multiple estimation methods into the distributionally robust optimization (DRO) framework. At the core of our approach is an innovative ambiguity set defined as the intersection of two Wasserstein balls, with their centers constructed using appropriate nonparametric and parametric estimators. On the computational side, we reformulate the IW-DRO problem as a tractable convex program and develop an approximate algorithm tailored for large-scale problems to enhance computational efficiency. From a theoretical perspective, we demonstrate that IW-DRO achieves superior performance compared to single Wasserstein-ball DRO models. We further establish performance guarantees by analyzing the coverage of the intersection ambiguity set and the measure concentration of both estimators under the Wasserstein distance. Notably, we derive a finite-sample concentration result for the Nadaraya-Watson kernel estimator under covariate shift. The proposed IW-DRO framework offers practical value for decision-makers operating in uncertain environments affected by covariate shifts.
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