Contextual Optimization under Covariate Shift: A Robust Approach by Intersecting Wasserstein Balls

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.