Tikhonov Regularization is Optimal Transport Robust under Martingale Constraints

Jiajin Li, Sirui Lin, Jose Blanchet, Viet Anh Nguyen

Keywords: Distributionally Robust Optimization, Optimal Transport, Tikhonov Regularization

Abstract: Distributionally robust optimization (DRO) 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 provide a unified viewpoint to a class of existing robust methods but also lead to new regularization tools. To realize these novel tools, provably efficient 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.

TL;DR: In this paper, we find that Tikhonov regularization is exactly distributionally robust in an optimal transport sense.