We revisit the question of characterizing the convergence rate of plug-in estimators of optimal transport costs. It is well known that an empirical measure comprising independent samples from an absolutely continuous distribution 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-in estimators of many optimal transport costs converge at this same rate. However, we show that when the cost is smooth, this analysis is loose: plug-in estimators based on empirical measures converge quadratically faster, at the rate $n^{-2/d}$. As a corollary, we show that the Wasserstein distance between two distributions is significantly easier to estimate when the measures are well-separated. We also prove lower bounds, showing not only that our analysis of the plug-in estimator is tight, but also that no other estimator can enjoy significantly 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 a byproduct of our proofs, we derive $L^\infty$ estimates on the displacement induced by the optimal coupling between any two measures satisfying suitable concentration and anticoncentration conditions, for a wide range of cost functions.

翻译：我们重新审视了最优输运代价的插件估计量收敛速率的刻画问题。众所周知，由来自 $\mathbb{R}^d$ 上绝对连续分布的独立样本构成的经验测度，在Wasserstein距离下以速率 $n^{-1/d}$ 收敛于该分布，这可用于证明许多最优输运代价的插件估计量以相同速率收敛。然而，我们证明当代价光滑时，这一分析是松散的：基于经验测度的插件估计量以速率 $n^{-2/d}$ 二次方更快地收敛。作为推论，我们表明当两个分布充分分离时，它们之间的Wasserstein距离显著更易估计。我们还证明了下界，这不仅表明我们的插件估计量分析是紧的，还表明没有任何其他估计量能在所有测度对上均匀地享有显著更快的收敛速率。我们的证明依赖于经验过程理论，基于对局部Lipschitz和半凹函数的$L^2$覆盖数的严格控制。作为证明的副产品，我们推导了在满足适当浓度与反浓度条件的任意两个测度之间，由最优耦合引起的位移的$L^\infty$估计，该结果适用于广泛的一类代价函数。