We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $\pi_{ t}$ is shown to satisfy $H(\pi_{ t}|\pi_{ *})+H(\pi_{ *}|\pi_{ t})=O(t^{ -1})$ where $H$ denotes relative entropy and $\pi_{ *}$ the optimal coupling. This holds for a large class of cost functions and marginals, including quadratic cost with subgaussian marginals. We also obtain the rate $O(t^{ -1})$ for the dual suboptimality and $O(t^{ -2})$ for the marginal entropies. More precisely, we derive non-asymptotic bounds, and in contrast to previous results on linear convergence that are limited to bounded costs, our estimates do not deteriorate exponentially with the regularization parameter. We also obtain a stability result for $\pi_{ *}$ as a function of the marginals, quantified in relative entropy.
Categories: math.OC math.AP math.PR