On the Convergence Rate of Sinkhorn's Algorithm

Abstract: We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $\pi_{t}$ is shown to satisfy $$H(\pi_{t}|\pi_{<em>})+H(\pi_{</em>}|\pi_{t})=O(t<sup>{-1})$$ where $H$ denotes relative entropy and $\pi_{<em>}$ 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 $O(t<sup>{-1})$ for the dual suboptimality and $O(t<sup>{-2})$ 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 $\pi_{</em>}$ as a function of the marginals, quantified in relative entropy.