On the Convergence Rate of Sinkhorn's Algorithm

We study Sinkhorn’s algorithm for solving the entropically regularized optimal transport problem. Its iterate ${\pi }_t$ is shown to satisfy $H({\pi }_t\mathrm{\mid }{\pi }_{\ast })+H({\pi }_{\ast }\mathrm{\mid }{\pi }_t)=O(t^{-1})$ where $H$ denotes relative entropy and ${\pi }_{\ast }$ 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 ${\pi }_{\ast }$ as a function of the marginals, quantified in relative entropy.