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

2024-12-12

AI Summary

This paper investigates the stability and exponential convergence of the Sinkhorn algorithm for entropy-regularized optimal transport. Focusing on quadratic cost, it establishes a Wasserstein stability theory based on semi-concavity assumptions, yielding the first global exponential convergence guarantee under log-concave marginals. It derives sharp convergence rate bounds with linear dependence on the regularization parameter. The analysis is extended to novel non-compact, unbounded settings—including Riemannian manifolds, elastic costs, and light-tailed marginals. Methodologically, the work integrates semi-concavity analysis, uniform upper bounds on the Hessian of Sinkhorn potentials, and refined Wasserstein distance estimation. Collectively, this provides the first unified exponential convergence guarantee for a broad class of generalized cost functions and marginal distributions. The derived rates improve upon prior results, significantly expanding the theoretical applicability of the Sinkhorn algorithm.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study stability of optimizers and convergence of Sinkhorn's algorithm in the framework of entropic optimal transport. We show entropic stability for optimal plans in terms of the Wasserstein distance between their marginals under a semiconcavity assumption on the sum of the cost and one of the two entropic potentials. 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. 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. Other interesting new applications include subspace elastic costs [Cuturi et al. PMLR 202(2023)], 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 unbounded Lipschitz costs, and compact Riemannian manifolds.

Problem

• Studying stability of entropic optimal transport optimizers and Sinkhorn convergence
• Establishing exponential convergence under semiconcavity without bounded cost
• Extending convergence results to various marginals and cost structures

Innovation

• Using semiconcavity to analyze Sinkhorn algorithm stability
• Controlling Hessians of potentials for exponential convergence
• Establishing optimal rates with linear regularization dependence