On the Convergence Rate of Sinkhorn's Algorithm

Promit Ghosal; Marcel Nutz

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arXiv · 2212.06000

On the Convergence Rate of Sinkhorn's Algorithm

Promit Ghosal·Marcel Nutz

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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.

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Promit Ghosal, Marcel Nutz. 2022-12-12. On the Convergence Rate of Sinkhorn's Algorithm. https://arxiv.org/abs/2212.06000

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