Network Consensus in the Wasserstein Metric Space of Probability Measures

Abstract

Distributed consensus in the Wasserstein metric space of probability measures on the real line is introduced in this work. Convergence of each agent's measure to a common measure is proven under a weak network connectivity condition. The common measure reached at each agent is one minimizing a weighted sum of its Wasserstein distance to all initial agent measures. This measure is known as the Wasserstein barycenter. Special cases involving Gaussian measures, empirical measures, and time-invariant network topologies are considered, where convergence rates and average-consensus results are given. This work has possible applicability in computer vision, machine learning, clustering, and estimation.

Distributed Wasserstein Barycenters via Displacement Interpolation

“…Our article is more in line with the spirit of [8] and [9], in the sense that we propose a theoretical formulation and analysis that prove how to generate Wasserstein barycenters from distributed computations. We do not propose specific designs of numerical solvers for the local computations of the agents, as it is instead performed in [23] and [36].…”

“…Moreover, our framework provides convergence guarantees for the computation of the barycenter independently from the numerical implementation of the local computations. We also remark that work [9] is different from ours because: (i) it only focuses on measures on the real line R, while we consider R d , d ≥ 1; (ii) its underlying communication graph is undirected at all time-steps and its changes are deterministic; (iii) it considers local computations of the full Wasserstein barycenter between agent and its neighbors; (iv) we present sufficient conditions for the computation of the standard Wasserstein barycenter.…”

“…Specifically, we contribute to the fields of randomized consensus algorithms-e.g., see [12,Ch. 13] and references therein-and of consensus in spaces other than the Euclidean space-e.g., see [9], [26], and [33]. Moreover, our work contributes to the field of opinion dynamics, where classic opinion models also use stochastic asynchronous pairwise interactions [1], [22].…”

“…To the best of our knowledge, there is only recent and growing literature on distributed algorithms for Wasserstein barycenters. The idea of computing Wasserstein barycenters in a distributed way was first pioneered by Bishop and Doucet in their work [8] and its very recent extension [9]. Their work formally shows consensus toward the Wasserstein barycenter of the agents' initial measures.…”

“…In order to compute such consensus, each agent needs to fully compute the Wasserstein barycenter resulting from its own measure and the measures from all its neighbors at each iteration according to some time-varying graph. The work [9] focuses on the case of probability measures on the real line, with the communication between agents being deterministic but flexible enough to consider both synchronous and asynchronous deterministic updating. It assumes that agents are connected by an undirected graph.…”