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A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm

Alberto Chiarini, Giovanni Conforti, Giacomo Greco, Luca Tamanini

TL;DR

The paper develops a semiconcavity-based framework for stability of entropic OT plans and for exponential convergence of Sinkhorn's algorithm in unbounded settings. By showing that a $\Lambda$-semiconcavity bound on the entropic potential $c(x,\cdot)+\psi_ε^{\nu}(\cdot)$ yields a KL stability bound $\mathrm{KL}(\pi_ε^{\rho\mu}|\pi_ε^{\rho\nu}) \le \mathrm{KL}(\mu|\nu) + \frac{\Lambda}{2ε}\mathbf{W}_2^2(\mu,\nu)$, the authors couple this with Talagrand-type transport inequalities to obtain geometric convergence of Sinkhorn iterates. The approach extends to $(\Lambda,\omega)$-semiconcavity and generalized TI, enabling exponential convergence across a wide range of marginals and costs, including log-concave, weakly log-concave, light-tailed, anisotropic quadratic, Lipschitz, and manifold settings, often with rates linear in the regularization parameter $ε$. The results sharpen previous bounds by relaxing Hessian-boundedness requirements and providing explicit, sometimes sharp, dependence on problem data, tail behavior, and geometric structure. The work thus broadens the applicability of entropic OT in unbounded domains and informs practical performance of Sinkhorn-type algorithms in diverse contexts. Overall, the paper advances both theory and potential applications of entropic OT by linking semiconcavity propagation to stability and fast convergence.

Abstract

We study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter, based on semiconcavity propagation results. Moreover, the convergence rate has a linear dependence on the regularization: this behavior is sharp and had only been previously obtained for compact distributions arXiv:2407.01202. These optimal rates are also established in situations where one of the two marginals does not have subgaussian tails. Other interesting new applications include subspace elastic costs, weakly log-concave marginals, marginals with light tails (where, under reinforced assumptions, we manage to improve the rates obtained in arXiv:2311.04041), the case of Lipschitz costs with bounded Hessian, and compact Riemannian manifolds.

A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm

TL;DR

The paper develops a semiconcavity-based framework for stability of entropic OT plans and for exponential convergence of Sinkhorn's algorithm in unbounded settings. By showing that a -semiconcavity bound on the entropic potential yields a KL stability bound , the authors couple this with Talagrand-type transport inequalities to obtain geometric convergence of Sinkhorn iterates. The approach extends to -semiconcavity and generalized TI, enabling exponential convergence across a wide range of marginals and costs, including log-concave, weakly log-concave, light-tailed, anisotropic quadratic, Lipschitz, and manifold settings, often with rates linear in the regularization parameter . The results sharpen previous bounds by relaxing Hessian-boundedness requirements and providing explicit, sometimes sharp, dependence on problem data, tail behavior, and geometric structure. The work thus broadens the applicability of entropic OT in unbounded domains and informs practical performance of Sinkhorn-type algorithms in diverse contexts. Overall, the paper advances both theory and potential applications of entropic OT by linking semiconcavity propagation to stability and fast convergence.

Abstract

We study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter, based on semiconcavity propagation results. Moreover, the convergence rate has a linear dependence on the regularization: this behavior is sharp and had only been previously obtained for compact distributions arXiv:2407.01202. These optimal rates are also established in situations where one of the two marginals does not have subgaussian tails. Other interesting new applications include subspace elastic costs, weakly log-concave marginals, marginals with light tails (where, under reinforced assumptions, we manage to improve the rates obtained in arXiv:2311.04041), the case of Lipschitz costs with bounded Hessian, and compact Riemannian manifolds.

Paper Structure

This paper contains 31 sections, 25 theorems, 208 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background on EOT and Sinkhorn algorithm
  3. Entropic stability for optimal plans
  4. Exponential convergence of Sinkhorn's algorithm
  5. Examples of application
  6. A $(\Lambda,\omega)$-semiconcavity example
  7. Literature review: Sinkhorn's algorithm
  8. A more precise comparison for the Euclidean quadratic cost
  9. Proofs
  10. Preliminaries
  11. Proof of the main results
  12. Proof of Proposition \ref{['prop:anisotropic:cost']}
  13. Log-concavity of $\rho$
  14. Log-concavity of $\rho$ and log-semiconvexity of $\nu$
  15. Marginal $\rho$ weakly log-concave
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let ass:differentiability hold and $\pi_\varepsilon^{\rho\nu},\,\pi_\varepsilon^{\rho\mu}$ be the unique optimizers in EOT for the set of marginals $(\rho,\nu)$ and $(\rho,\mu)$. If there exists $\Lambda>0$ such that is $\Lambda$-semiconcave uniformly in $x\in\mathop{\mathrm{supp}}\nolimits(\rho)$, then

Theorems & Definitions (42)

  • Theorem 1.1: KL stability of optimal plans
  • Theorem 1.2: KL stability of optimal plans generalized
  • Theorem 1.3: Exponential convergence of Sinkhorn's algorithm
  • Theorem 1.4: Exponential convergence in $(\Lambda,\omega)$-semiconcave settings
  • Proposition 1: Anisotropic quadratic costs and (weakly) log-concave marginals
  • Remark 1
  • Proposition 2: $\rho$ with light tails
  • Proposition 3: Lipschitz costs
  • Proposition 4: $\mathop{\mathrm{supp}}\nolimits(\rho)$ compact
  • Proposition 5: Riemannian sphere
  • ...and 32 more