Distributionally robust optimization has been shown to offer a principled way
to regularize learning models. In this paper, we find that Tikhonov
regularization is distributionally robust in an optimal transport sense (i.e.,
if an adversary chooses distributions in a suitable optimal transport
neighborhood of the empirical measure), provided that suitable martingale
constraints are also imposed. Further, we introduce a relaxation of the
martingale constraints which not only provides a unified viewpoint to a class
of existing robust methods but also leads to new regularization tools. To
realize these novel tools, tractable computational algorithms are proposed. As
a byproduct, the strong duality theorem proved in this paper can be potentially
applied to other problems of independent interest.Comment: Accepted by NeurIPS 202