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).

翻译：我们研究基于最优运输的分布鲁棒优化问题，其中虚构的对抗者（通常被视为自然）可以通过在有限运输成本下重塑预设参考分布，来选择不确定问题参数的分布。在此框架下，我们证明即使运输成本函数不是（某次幂的）度量，鲁棒化也与各种形式的变异正则化和Lipschitz正则化密切相关。我们还推导了决策者与自然之间纳什均衡存在性与可计算性的条件，并通过数值实验表明，自然的纳什策略可视为一种支持于极具欺骗性的对抗样本的分布。最后，我们识别出若干实际相关的基于最优运输的分布鲁棒优化问题类别，这些问题即使损失函数或运输成本函数非凸（但不同时成立），也能通过高效的梯度下降算法求解。