Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

Stephan Eckstein, Marcel Nutz

Abstract: We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions, including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in the Wasserstein sense, including for quadratic cost. Two techniques are presented: the first compares an optimal coupling with its so-called shadow, which is a coupling induced on other marginals by an explicit construction, and the second transforms one set of marginals by a change of coordinates and thus reduces the comparison of differing marginals to the comparison of differing cost functions under the same marginals.