Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

Hugo Malamut, Maxime Sylvestre

TL;DR

This work disentangles the impact of the cost term and entropy in entropy-regularized optimal transport as the regularization vanishes. By leveraging the Schrödinger problem, Benamou–Brenier dynamics, and Minty’s quadratic detachment, it provides precise asymptotics: with quadratic costs and suitable moment or entropy conditions, the entropy scales as −(d/2) ln ε while the suboptimality in cost scales linearly in ε, and W_2 distances shrink at rates tied to ε or √ε depending on regularity. For infinitesimally twisted costs, the authors establish analogous rate results and derive lower bounds on entropy and Wasserstein distances, clarifying how regularity and marginal concentration affect convergence. The results collectively offer a detailed, rate-aware decomposition of entropy and cost effects, with implications for understanding and improving entropic OT approximations and algorithms. Practically, these findings inform the accuracy of entropic regularization-based OT computations in high-dimensional settings where marginals have finite entropy and regularity.

Abstract

We study the convergence of the transport plans $γ_ε$ towards $γ_0$ as well as the cost of the entropy-regularized optimal transport $(c,γ_ε)$ towards $(c,γ_0)$ as the regularization parameter $ε$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(γ_ε,γ_0)$ is asymptotically greater than $C\sqrtε$ and the suboptimality $(c,γ_ε)-(c,γ_0)$ is of order $ε$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+δ$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(γ_ε,γ_0)$ converges to $0$ at rate $\sqrtε$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,γ_ε)-(c,γ_0) \sim dε/2$ and we provide a companion expansion of $H(γ_ε)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.

Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

TL;DR

This work disentangles the impact of the cost term and entropy in entropy-regularized optimal transport as the regularization vanishes. By leveraging the Schrödinger problem, Benamou–Brenier dynamics, and Minty’s quadratic detachment, it provides precise asymptotics: with quadratic costs and suitable moment or entropy conditions, the entropy scales as −(d/2) ln ε while the suboptimality in cost scales linearly in ε, and W_2 distances shrink at rates tied to ε or √ε depending on regularity. For infinitesimally twisted costs, the authors establish analogous rate results and derive lower bounds on entropy and Wasserstein distances, clarifying how regularity and marginal concentration affect convergence. The results collectively offer a detailed, rate-aware decomposition of entropy and cost effects, with implications for understanding and improving entropic OT approximations and algorithms. Practically, these findings inform the accuracy of entropic regularization-based OT computations in high-dimensional settings where marginals have finite entropy and regularity.

Abstract

We study the convergence of the transport plans towards as well as the cost of the entropy-regularized optimal transport towards as the regularization parameter vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance is asymptotically greater than and the suboptimality is of order . In the quadratic cost case the compactness assumption is relaxed into a moment of order assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance converges to at rate . Finally, if in addition the marginals have finite Fisher information, we prove and we provide a companion expansion of . These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.
Paper Structure (14 sections, 21 theorems, 102 equations)

This paper contains 14 sections, 21 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Definitions and assumptions
  4. Exact asymptotics from the Schrödinger problem
  5. Schrödinger problem and Benamou-Brenier formulation
  6. A precise expansion of $H(\gamma_\varepsilon)$ and $(c,\gamma_\varepsilon)$
  7. Rephrasing the results with different functionals
  8. Rates for the quadratic cost
  9. Lower bounds on the entropy
  10. Rates of convergence
  11. Rates for infinitesimally twisted costs
  12. Lower bound on the entropy
  13. Rates of convergence of $OT_\varepsilon$
  14. Lower bound on $W_2(\gamma_\varepsilon,\gamma_0)$

Key Result

Theorem 1

Suppose that the cost is quadratic, that is $c(x,y) = \frac{1}{2}\Vert x-y \Vert^2$. Further assume that $I(\mu_i)< \infty$ and $\text{supp}(\mu_i)$ compact. Then and

Theorems & Definitions (46)

  • Theorem : Theorem \ref{['thm:expansions']}
  • Theorem : Theorem \ref{['thm:rate-cost-quadratic']}
  • Theorem : Theorem \ref{['thm:general-rates']}, Theorem \ref{['thm:gen-mino-wass-any-coupling']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 36 more