Nash Equilibria, Regularization, and Computation in Optimal Transport-Based Distributionally Robust Optimization

We study optimal transport-based distributionally robust optimization problems in which a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision maker and nature, and we demonstrate numerically that nature’s Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function is nonconvex (but not both at the same time).

Funding: This research was supported by the Swiss National Science Foundation under the National Centres of Competence in Research Automation [Grant 51NF40_180545] and under Early Postdoc.Mobility Fellowships awarded to the first and second authors [Grants P2ELP2_195149 and P500PT_222215, respectively].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2023.0138.