Wasserstein Distributionally Robust Optimization and Variation Regularization
This work addresses the need for robust optimization methods in machine learning and operations research by providing a foundational theory that relaxes restrictive assumptions, though it is incremental in extending existing regularization connections.
The paper develops a general theory linking Wasserstein distributionally robust optimization to variation regularization, applicable to non-convex, non-smooth losses and non-Euclidean spaces, and uses this to derive new generalization guarantees for adversarial robust learning.
Wasserstein distributionally robust optimization (DRO) has recently achieved empirical success for various applications in operations research and machine learning, owing partly to its regularization effect. Although connection between Wasserstein DRO and regularization has been established in several settings, existing results often require restrictive assumptions, such as smoothness or convexity, that are not satisfied for many problems. In this paper, we develop a general theory on the variation regularization effect of the Wasserstein DRO - a new form of regularization that generalizes total-variation regularization, Lipschitz regularization and gradient regularization. Our results cover possibly non-convex and non-smooth losses and losses on non-Euclidean spaces. Examples include multi-item newsvendor, portfolio selection, linear prediction, neural networks, manifold learning, and intensity estimation for Poisson processes, etc. As an application of our theory of variation regularization, we derive new generalization guarantees for adversarial robust learning.