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：我们研究的是最佳的基于运输的分布强力优化问题,因为假的对手(通常被设想为自然)可以选择不确定问题参数的分布方式,以有限的运输成本重塑规定的参考分布。在这个框架内,我们表明,稳健化与各种形式的变异和利普施奇茨正规化密切相关,即使运输成本功能不能(某种能力)作为衡量标准。我们还为纳什平衡在决策者和自然之间的存在和可乘性创造了条件,我们从数字上证明,自然的纳什战略可以被视为一种分布方式,这种分配方式得到明显欺骗性的对抗性样本的支持。最后,我们确定了可以使用高效梯度下降算法解决的基于运输的最佳分布稳健的优化问题,即使损失功能或运输成本功能不是凝固(但不能同时同时) 。</s>