  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.
