This was part of Distributed Solutions to Complex Societal Problems Reunion Workshop

## On the Convergence Rate of Sinkhorn’s Algorithm

Marcel Nutz, Columbia University
Tuesday, February 21, 2023

Abstract: We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $pi_{t}$ is shown to satisfy $H(pi_{t}|pi_{*})+H(pi_{*}|pi_{t})=O(t^{-1})$ 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. Joint work with Promit Ghosal (MIT), arXiv:2212.06000.