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Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Tudor Manole, Jonathan Niles-Weed

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 $\mathbb{R}^d$ 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.

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 converges to that distribution at the rate 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 . 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 covering numbers for locally Lipschitz and semi-concave functions. As a byproduct of our proofs, we derive 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.

Paper Structure

This paper contains 34 sections, 25 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Example: Location Families
  3. Our Contributions
  4. Related Work
  5. Notation and Further Background on the Optimal Transport Problem
  6. Upper Bounds for Compactly Supported Measures
  7. Proof of Theorem \ref{['thm:ub_general']}
  8. Case 1: $\alpha = 2$
  9. Case 2: $1 < \alpha < 2$
  10. Case 3: $0 < \alpha \leq 1$
  11. Upper Bounds for Unbounded Measures under Tail Conditions
  12. Proof of Theorem \ref{['thm:main_unbounded']}
  13. Proof of Theorem \ref{['thm:coupling_quantitative']}
  14. Lower Bounds
  15. Lower Bounds for the Empirical Optimal Transport Cost
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\mu,\nu \in \calP(\bbR^d)$ and $p > 1$. Let $\nu$ be a $\sigma^2$-sub-Gaussian measure boucheron2013 and $\mu$ have finite $p$-th moment, and assume there exist constants $c_1,c_2 > 0$ such that $\mu(B_{x,1}) \geq c_1 \exp(-c_2 \norm x^2)$ for all $x \in \bbR^d$. Then, for any optimal coupling In particular, if there exists an optimal transport map $T$ from $\mu$ to $\nu$ with respect to $c_

Theorems & Definitions (30)

  • Theorem : Informal
  • Lemma 1
  • Theorem 2
  • Corollary 3: Powers of $\ell_r$ Norms
  • Corollary 4: Wasserstein Distances
  • proof
  • Lemma 5
  • Lemma 6: vonluxburg2004
  • Lemma 7: bronshtein1976
  • Lemma 8
  • ...and 20 more