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

Abstract

We study the convergence of the transport plans ${\hat{I}}_{\hat{I}}^3\mu$ towards ${\hat{I}}_0^3$ as well as the cost of the entropy-regularized optimal transport $(c,{\hat{I}}_{\hat{I}}^3\mu )$ towards $(c,{\hat{I}}_0^3)$ as the regularization parameter $\hat{I}\mu$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2({\hat{I}}_{\hat{I}}^3\mu ,{\hat{I}}_0^3)$ is asymptotically greater than $C\sqrt{\hat{I}}\mu$ and the suboptimality $(c,{\hat{I}}_{\hat{I}}^3\mu )-(c,{\hat{I}}_0^3)$ is of order $\hat{I}\mu$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+\hat{I}´$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2({\hat{I}}_{\hat{I}}^3\mu ,{\hat{I}}_0^3)$ converges to $0$ at rate $\sqrt{\hat{I}}\mu$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,{\hat{I}}_{\hat{I}}^3\mu )-(c,{\hat{I}}_0^3)\sim d\hat{I}\mu /2$ and we provide a companion expansion of $H({\hat{I}}_{\hat{I}}^3\mu )$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.