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We revisit the question of characterizing the convergence rate of plug-inestimators of optimal transport costs. It is well known that an empiricalmeasure comprising independent samples from an absolutely continuousdistribution on $\mathbb{R}^d$ converges to that distribution at the rate$n^{-1/d}$ in Wasserstein distance, which can be used to prove that plug-inestimators of many optimal transport costs converge at this same rate. However,we show that when the cost is smooth, this analysis is loose: plug-inestimators based on empirical measures converge quadratically faster, at therate $n^{-2/d}$. As a corollary, we show that the Wasserstein distance betweentwo distributions is significantly easier to estimate when the measures are farapart. We also prove lower bounds, showing not only that our analysis of theplug-in estimator is tight, but also that no other estimator can enjoysignificantly faster rates of convergence uniformly over all pairs of measures.Our proofs rely on empirical process theory arguments based on tight control of$L^2$ covering numbers for locally Lipschitz and semi-concave functions. As abyproduct of our proofs, we derive $L^\infty$ estimates on the displacementinduced by the optimal coupling between any two measures satisfying suitablemoment conditions, for a wide range of cost functions.

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs