Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters

TL;DR

This paper introduces a novel accelerated primal-dual stochastic gradient method for decentralized computation of Wasserstein barycenters, achieving faster convergence with explicit complexity bounds in distributed settings.

Contribution

It develops a new algorithm for decentralized Wasserstein barycenter computation and provides non-asymptotic complexity analysis, advancing distributed optimal transport methods.

Findings

The proposed method accelerates convergence in decentralized Wasserstein barycenter computation.

Explicit non-asymptotic complexity bounds are derived for the algorithm.

The approach effectively handles stochastic convex optimization with linear constraints.

Abstract

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decentralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm.