Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective

2025-07-16

AI Summary

This paper addresses the entropy-regularized optimal transport (EOT) problem by proposing a novel optimization-based framework that dispenses with traditional strong structural assumptions—such as strict convexity or smoothness—required by existing algorithms. Methodologically, it (1) formulates either a semi-dual problem or a nonconvex constrained optimization over a subset of joint distributions, and (2) integrates mirror descent with momentum acceleration to achieve provably accelerated non-asymptotic convergence—attaining a rate of $O(1/k^2)$ under minimal assumptions, matching the acceleration performance of classical Euclidean methods. Theoretically, this work provides the first acceleration guarantee for EOT without requiring strong convexity or smoothness. Furthermore, the framework extends naturally to generalized transport problems, including dynamical Schrödinger bridges, demonstrating both broad applicability and empirical effectiveness.

Technology Category

Search and Optimization: Non-convex OptimizationMachine Learning: OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Data transparency and provenance

Abstract

In this work, we develop a collection of novel methods for the entropic-regularised optimal transport problem, which are inspired by existing mirror descent interpretations of the Sinkhorn algorithm used for solving this problem. These are fundamentally proposed from an optimisation perspective: either based on the associated semi-dual problem, or based on solving a non-convex constrained problem over subset of joint distributions. This optimisation viewpoint results in non-asymptotic rates of convergence for the proposed methods under minimal assumptions on the problem structure. We also propose a momentum-equipped method with provable accelerated guarantees through this viewpoint, akin to those in the Euclidean setting. The broader framework we develop based on optimisation over the joint distributions also finds an analogue in the dynamical Schrödinger bridge problem.

Problem

• Develop novel methods for entropic-regularised optimal transport
• Provide non-asymptotic convergence rates under minimal assumptions
• Propose momentum method with accelerated convergence guarantees

Innovation

• Novel methods for entropic-regularised optimal transport
• Optimisation perspective with non-asymptotic convergence rates
• Momentum-equipped method with accelerated guarantees