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

TL;DR

This paper investigates the stability of entropically regularized optimal transport with respect to marginals, establishing Lipschitz and Hölder continuity results, and demonstrates convergence of Sinkhorn's algorithm under general conditions.

Contribution

It provides new stability bounds for regularized optimal transport and proves convergence of Sinkhorn's algorithm in Wasserstein distance, including for quadratic costs.

Findings

Lipschitz continuity of the value function.

Hölder continuity of the optimal coupling.

Convergence of Sinkhorn's algorithm in Wasserstein sense.

Abstract

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older 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 Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling induced on other marginals by an explicit construction. 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.