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