Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

TL;DR

This paper analyzes how entropy-regularized optimal transport plans and costs converge to their unregularized counterparts as the regularization parameter approaches zero, revealing precise rates and conditions for convergence.

Contribution

It provides new asymptotic convergence rates for regularized optimal transport plans and costs, disentangling the effects of cost and entropy regularization under various assumptions.

Findings

W2 distance between plans is asymptotically greater than C√ε.

Suboptimality is of order ε as regularization vanishes.

Convergence rate of W2 distance is √ε under Lipschitz transport maps.

Abstract

We study the convergence of the transport plans $\gamma_\epsilon$ towards $\gamma_0$ as well as the cost of the entropy-regularized optimal transport $(c,\gamma_\epsilon)$ towards $(c,\gamma_0)$ as the regularization parameter $\epsilon$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(\gamma_\epsilon,\gamma_0)$ is asymptotically greater than $C\sqrt{\epsilon}$ and the suboptimality $(c,\gamma_\epsilon)-(c,\gamma_0)$ is of order $\epsilon$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+\delta$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(\gamma_\epsilon,\gamma_0)$ converges to $0$ at rate $\sqrt{\epsilon}$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,\gamma_\epsilon)-(c,\gamma_0) \sim d\epsilon/2$ and we provide a companion expansion of $H(\gamma_\epsilon)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.