Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective

TL;DR

This paper introduces new optimization-based algorithms for entropic optimal transport, providing convergence guarantees and acceleration techniques, with potential applications to Schrödinger bridge problems.

Contribution

It develops novel optimization algorithms for entropic optimal transport inspired by mirror descent, with convergence rates and acceleration guarantees from an optimization perspective.

Findings

Non-asymptotic convergence rates established

Proposed momentum method achieves accelerated guarantees

Framework applicable to Schrödinger bridge problems

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 Schr\"{o}dinger bridge problem.