On the Convergence Rate of Sinkhorn's Algorithm

TL;DR

This paper analyzes the convergence rate of Sinkhorn's algorithm for entropically regularized optimal transport, providing non-asymptotic bounds and stability results that extend previous linear convergence findings.

Contribution

It establishes explicit convergence rates for Sinkhorn's algorithm under broad conditions, including unbounded costs, and introduces stability bounds for the optimal coupling.

Findings

Convergence rate of O(t^{-1}) for the relative entropy between iterates and the optimum.

Rate of O(t^{-2}) for marginal entropies.

Non-asymptotic bounds that do not worsen exponentially with regularization.

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.