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

We study the convergence of the transport plans γε towards γ0 as well as the cost of the entropy-regularized optimal transport (c,γε) towards (c,γ0) as the regularization parameter ε vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance W2(γε,γ0) is asymptotically greater than C√ε and the suboptimality (c,γε)-(c,γ0) is of order ε. In the quadratic cost case the compactness assumption is relaxed into a moment of order 2+δ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance W2(γε,γ0) converges to 0 at rate √ε. Finally, if in addition the marginals have finite Fisher information, we prove (c,γε)-(c,γ0) ~ dε/2 and we provide a companion expansion of H(γε). These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.