Finite-sample bounds for regularized optimal transport

Abstract

We study the sample complexity of regularized optimal transport for general convex regularizations including the Kullback--Leibler divergence and $L^p$ penalties. Our main results are non-asymptotic bias and variance bounds for the empirical cost, with explicit dependence on the regularization parameter and on the intrinsic dimension of the marginals. Our approach simultaneously improves, unifies, and extends existing finite-sample bounds. In particular, we improve the state of the art for entropic optimal transport, and we obtain the first fully quantitative results for $L^p$ regularization with $1<p<\infty$. For the quadratic transport cost, we deduce that quadratically regularized optimal transport (i.e., $L^2$ regularization) estimates the unregularized optimal transport cost at rate $n^{-2/(d+4)}$, the fastest non-asymptotic rate currently available for any estimator based on regularized optimal transport.