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Wasserstein distributionally robust optimization (WDRO) attempts to learn amodel that minimizes the local worst-case risk in the vicinity of the empiricaldata distribution defined by Wasserstein ball. While WDRO has receivedattention as a promising tool for inference since its introduction, itstheoretical understanding has not been fully matured. Gao et al. (2017)proposed a minimizer based on a tractable approximation of the local worst-caserisk, but without showing risk consistency. In this paper, we propose aminimizer based on a novel approximation theorem and provide the correspondingrisk consistency results. Furthermore, we develop WDRO inference for locallyperturbed data that include the Mixup (Zhang et al., 2017) as a special case.We show that our approximation and risk consistency results naturally extend tothe cases when data are locally perturbed. Numerical experiments demonstraterobustness of the proposed method using image classification datasets. Ourresults show that the proposed method achieves significantly higher accuracythan baseline models on noisy datasets.

Principled learning method for Wasserstein distributionally robust optimization with local perturbations