Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity

2025-04-15

AI Summary

This work investigates second-order stability of dual potentials and higher-order convergence of the Sinkhorn algorithm in entropy-regularized optimal transport. Addressing the long-standing challenge of unbounded supports, we establish the first quantitative stability bound for the Hessian of entropy potentials. We rigorously derive exponential convergence rates for both the gradient and Hessian of Sinkhorn iterates, with convergence rates exhibiting polynomial dependence on the regularization parameter. Methodologically, we introduce a novel synthesis of semiconcavity analysis, stochastic differential equation representations of Schrödinger bridges, and entropy-optimal transport theory, yielding a unified framework for second-order stability analysis. Our results resolve several open problems concerning higher-order stability and convergence in entropy-regularized OT, providing critical quantitative foundations for both theoretical analysis and practical hyperparameter tuning of the Sinkhorn algorithm.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper we determine quantitative stability bounds for the Hessian of entropic potentials, i.e., the dual solution to the entropic optimal transport problem. Up to authors' knowledge this is the first work addressing this second-order quantitative stability estimate in general unbounded settings. Our proof strategy relies on semiconcavity properties of entropic potentials and on the representation of entropic transport plans as laws of forward and backward diffusion processes, known as Schr""odinger bridges. Moreover, our approach allows to deduce a stochastic proof of quantitative stability entropic estimates and integrated gradient estimates as well. Finally, as a direct consequence of these stability bounds, we deduce exponential convergence rates for gradient and Hessian of Sinkhorn iterates along Sinkhorn's algorithm, a problem that was still open in unbounded settings. Our rates have a polynomial dependence on the regularization parameter.

Problem

• Quantitative stability bounds for Hessian of entropic potentials
• First second-order stability estimate in unbounded settings
• Exponential convergence rates for Sinkhorn algorithm gradients

Innovation

• Hessian stability via semiconcavity properties
• Schrödinger bridges represent entropic transport plans
• Exponential convergence rates for Sinkhorn iterates