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.

本文提出了一种基于新近估算定理的WDRO（Wasserstein分布鲁棒优化）最小化器，通过实现该最小化器及提供相应的风险一致性结果，发现该方法成功应用于本地扰动数据的WDRO推断，同时对噪声数据如图像分类数据集表现出鲁棒性优异的效果。

具原则的学习方法：带有局部扰动的Wasserstein分布鲁棒优化