Accelerated Bregman Primal-Dual methods applied to Optimal Transport and Wasserstein Barycenter problems

TL;DR

This paper introduces accelerated Bregman primal-dual algorithms for optimal transport and Wasserstein barycenter problems, achieving state-of-the-art convergence rates and improved numerical stability through novel divergence measures.

Contribution

It extends hybrid primal-dual methods with linesearch to Bregman divergences, providing new algorithms with enhanced convergence and stability for OT and WB problems.

Findings

Achieves state-of-the-art convergence rates both theoretically and practically.

Introduces a Bregman divergence based on scaled entropy for numerical stability.

Demonstrates improved solutions with numerical experiments.

Abstract

This paper discusses the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman's kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.