Decentralized Convex Optimization on Time-Varying Networks with Application to Wasserstein Barycenters

TL;DR

This paper introduces a new distributed algorithm for Wasserstein barycenters that operates efficiently over time-varying networks, with proven convergence rates and demonstrated practical effectiveness.

Contribution

It develops the first first-order method for non-smooth dual-friendly distributed optimization on dynamic networks tailored to Wasserstein barycenters.

Findings

Proves accelerated convergence rates depending on network parameters.

Demonstrates algorithm efficiency on Wasserstein barycenter tasks.

Handles non-smooth optimization in dynamic network settings.

Abstract

Inspired by recent advances in distributed algorithms for approximating Wasserstein barycenters, we propose a novel distributed algorithm for this problem. The main novelty is that we consider time-varying computational networks, which are motivated by examples when only a subset of sensors can make an observation at each time step, and yet, the goal is to average signals (e.g., satellite pictures of some area) by approximating their barycenter. We embed this problem into a class of non-smooth dual-friendly distributed optimization problems over time-varying networks, and develop a first-order method for this class. We prove non-asymptotic accelerated in the sense of Nesterov convergence rates and explicitly characterise their dependence on the parameters of the network and its dynamics. In the experiments, we demonstrate the efficiency of the proposed algorithm when applied to the Wasserstein barycenter problem.