We study distributionally robust optimization (DRO) with Sinkhorn distance -- a variant of Wasserstein distance based on entropic regularization. We derive convex programming dual reformulation for general nominal distributions, transport costs, and loss functions. Compared with Wasserstein DRO, our proposed approach offers enhanced computational tractability for a broader class of loss functions, and the worst-case distribution exhibits greater plausibility in practical scenarios. To solve the dual reformulation, we develop a stochastic mirror descent algorithm with biased gradient oracles. Remarkably, this algorithm achieves near-optimal sample complexity for both smooth and nonsmooth loss functions, nearly matching the sample complexity of the Empirical Risk Minimization counterpart. Finally, we provide numerical examples using synthetic and real data to demonstrate its superior performance.

翻译：我们研究了基于Sinkhorn距离（一种基于熵正则化的Wasserstein距离变体）的分布鲁棒优化（DRO）。针对一般名义分布、传输代价和损失函数，我们推导了其凸规划对偶重构形式。与Wasserstein DRO相比，所提方法在更广泛的损失函数类别上具有更强的计算可处理性，且最坏情况分布在实际场景中表现出更高的合理性。为求解对偶重构问题，我们开发了一种基于有偏梯度oracle的随机镜像下降算法。值得注意的是，该算法在光滑与非光滑损失函数上均能达到接近最优的样本复杂度，几乎与经验风险最小化方法的样本复杂度相匹配。最后，我们通过合成数据与真实数据的数值实验展示了其优越性能。