On the Convergence Rate of Sinkhorn's Algorithm

TL;DR

This work establishes non-asymptotic convergence rates for Sinkhorn's algorithm solving entropically regularized OT with potentially unbounded costs. It combines a two-step approach—moment control of dual iterates and a conjugate-function analysis—with a Bolley–Villani weighted CKP framework to prove $H(\pi_{*}|\pi_{t})+H(\pi_{t}|\pi_{*})=O(t^{-1})$, dual suboptimality $O(t^{-1})$, and marginal entropies $O(t^{-2})$, while providing stability of the optimal coupling to marginal perturbations. The results apply to broad cost classes, including quadratic costs with subgaussian marginals, and yield bounds that scale polynomially, not exponentially, with the regularization parameter $\varepsilon$. Additionally, the paper develops uniform bounds for c-conjugates/biconjugates to control the dual potentials, yielding practical, non-asymptotic rates that extend beyond bounded-cost settings and enhance theoretical understanding of Sinkhorn convergence in high dimensions.

Abstract

We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $π_{t}$ is shown to satisfy $H(π_{t}|π_{*})+H(π_{*}|π_{t})=O(t^{-1})$ where $H$ denotes relative entropy and $π_{*}$ 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 $π_{*}$ as a function of the marginals, quantified in relative entropy.

Theorems & Definitions (41)

• Example 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3: Stability
• Remark 3.4
• Remark 3.5
• Lemma 3.6
• proof