Distributionally robust optimization has emerged as an attractive way to train robust machine learning models, capturing data uncertainty and distribution shifts. Recent statistical analyses have proved that robust models built from Wasserstein ambiguity sets have nice generalization guarantees, breaking the curse of dimensionality. However, these results are obtained in specific cases, at the cost of approximations, or under assumptions difficult to verify in practice. In contrast, we establish, in this article, exact generalization guarantees that cover all practical cases, including any transport cost function and any loss function, potentially non-convex and nonsmooth. For instance, our result applies to deep learning, without requiring restrictive assumptions. We achieve this result through a novel proof technique that combines nonsmooth analysis rationale with classical concentration results. Our approach is general enough to extend to the recent versions of Wasserstein/Sinkhorn distributionally robust problems that involve (double) regularizations.

翻译：分布鲁棒优化已成为训练鲁棒机器学习模型的一种有吸引力方法，能够捕获数据不确定性和分布偏移。最近的统计分析表明，基于Wasserstein模糊集构建的鲁棒模型具有良好的泛化保证，突破了维度灾难。然而，这些结果是在特定情形下通过近似代价获得的，或基于实践中难以验证的假设。相比之下，本文建立了涵盖所有实际情形的精确泛化保证，包括任意传输成本函数和任意损失函数（可能非凸且非光滑）。例如，我们的结果适用于深度学习，无需强加限制性假设。我们通过一种结合非光滑分析原理与经典集中结果的新颖证明技术实现了这一成果。该方法具有足够通用性，可扩展至涉及（双重）正则化的最新Wasserstein/Sinkhorn分布鲁棒问题。