New Perspectives on Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization

2023-03-07

arXiv.org

AI Summary

This paper investigates risk-averse modeling in distributionally robust optimization (DRO) via optimal transport, framing the problem as a zero-sum game between a decision-maker and a “nature” adversary that perturbs a reference distribution within a budget of transportation cost. We propose a novel DRO framework with non-metric transportation costs and establish, for the first time, its exact equivalence to explicit total-variation or Lipschitz regularization. We prove existence and computability of Nash equilibria and show that nature’s optimal strategy concentrates on sparse, highly adversarial perturbations. Crucially, our framework enables efficient gradient-based optimization even under nonconvex loss functions—or nonconvex transportation costs—but not both simultaneously. Theoretical analysis and numerical experiments jointly demonstrate unified improvements in solution stability, computational tractability, and adversarial robustness.

Abstract

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).

Problem

• Robust Optimization
• Nash Equilibrium
• Stability under Uncertainty

Innovation

• Optimal Transport Theory
• Robust Optimization
• Gradient Descent Algorithm