Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization
This work addresses robust optimization problems for decision-makers facing uncertain parameters, offering incremental advances in regularization and computational methods.
The paper tackles optimal transport-based distributionally robust optimization by showing that robustification relates to regularization and establishing conditions for Nash equilibrium existence and computability, with numerical demonstrations of deceptive adversarial samples and identification of efficiently solvable problem classes.
We study optimal transport-based distributionally robust optimization problems where 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 are nonconvex (but not both at the same time).