A central question in statistical learning is to design algorithms that not only perform well on training data, but also generalize to new and unseen data. In this paper, we tackle this question by formulating a distributionally robust stochastic optimization (DRSO) problem, which seeks a solution that minimizes the worst-case expected loss over a family of distributions that are close to the empirical distribution in Wasserstein distances. We establish a connection between such Wasserstein DRSO and regularization. More precisely, we identify a broad class of loss functions, for which the Wasserstein DRSO is asymptotically equivalent to a regularization problem with a gradient-norm penalty. Such relation provides new interpretations for problems involving regularization, including a great number of statistical learning problems and discrete choice models (e.g. multinomial logit). The connection also suggests a principled way to regularize high-dimensional, non-convex problems, which is demonstrated the training of Wasserstein generative adversarial networks (WGANs) in deep learning.
Wasserstein distributionally robust stochastic optimization has been related to regularization by certain empirical gradient norms REF .
Distributional Robustness and Regularization in Statistical Learning