Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Abstract

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 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 far apart. 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 ∞ estimates on the displacement induced by the optimal coupling between any two measures satisfying suitable moment conditions, for a wide range of cost functions.

Limit distribution theory for smooth $p$-Wasserstein distances

“…The rest of the proof mirrors the latter half of the proof of Lemma 9 in [MNW21]. Since g ∈ L 1 (µ * γ σ ) and µ * γ σ is equivalent to the Lebesgue measure (i.e., µ * γ σ ≪ dx and dx ≪ µ * γ σ ), g(x) > −∞ for a.e.…”

“…The proof of Lemma 5.3 borrows ideas from Lemmas 9 and 10 and Theorem 11 in the recent work by [MNW21], which in turn build on [GM96,CF21]. Next, by Proposition 2 in [PW16], µ * γ σ has Lebesgue density f µ that is (c 1 , c 2 )regular with c 1 = 3/σ 2 and c 2 = 4 E µ [|X|]/σ 2 , i.e.,…”

“…This rate is known to be sharp in general [Dud69]. The recent work by [CRL + 20,MNW21] discovered that the rate can be improved under the alternative, namely,…”

Plugin Estimation of Smooth Optimal Transport Maps

“…Characterizing the convergence rate of the empirical measure under the Wasserstein distance is a classical problem (Dudley, 1969;Boissard and Le Gouic, 2014;Fournier and Guillin, 2015;Weed and Bach, 2019;Lei, 2020) which immediately leads to upper bounds on the convergence rate of the empirical plugin estimator of the Wasserstein distance. While such upper bounds are generally unimprovable (Liang, 2019;Niles-Weed and Rigollet, 2019), they have recently been sharpened by Chizat et al (2020) and Manole and Niles-Weed (2021) when P = Q, and we employ these results to bound the convergence rates of our empirical optimal transport map estimators in Sections 3.2 and 4.2. Though the empirical plugin estimator of the Wasserstein distance is minimax optimal up to polylogarithmic factors under no assumptions on P and Q, it becomes suboptimal when P and Q are assumed to have smooth densities.…”

Wasserstein Conditional Independence Testing

“…[BGV07,WB19]). Note also that if µ = ν, then the convergence of W p (µ n , ν n ) to W p (µ, ν) was recently shown to occur at a faster rate than O(n −1/d ) in general [CRL + 20,MNW21]. Now let's return our focus to (two-sample) independence testing for X and Y .…”