Network Consensus in the Wasserstein Space of Probability Measures Defined on Multi-Dimensional Euclidean Spaces
Pilgyu Jung, Yoon Mo Jung
TL;DR
The paper addresses consensus among agents whose states are probability measures in the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$, where curvature for $d>1$ complicates convexity-based analysis. It introduces a nonlinear update using the Wasserstein barycenter and leverages a Wasserstein Jensen inequality to obtain displacement-convexity-based estimates under joint network connectivity and a positive lower bound $\delta$ on weights. The main result shows that the scheme drives all agents to a common limit $\mu^*\in\mathcal{P}_2(\mathbb{R}^d)$ with $W_p(\mu_i(t),\mu^*)\to 0$ for all $p\in[1,2)$, and under stronger initial moment conditions, convergence in $W_2$ is guaranteed. This work extends one-dimensional consensus results to higher dimensions, enabling distributed aggregation of distributions and domain adaptation in transport-based geometric settings.
Abstract
The consensus problem -- achieving agreement among a network of agents -- is a central theme in both theory and applications. Recently, this problem has been extended from Euclidean spaces to the space of probability measures, where the natural notion of averaging is given by the Wasserstein barycenter. While prior work established convergence in one dimension, the case of higher dimensions poses additional challenges due to the curved geometry of Wasserstein space. In this paper, we develop a framework for analyzing such consensus algorithms by employing a Wasserstein version of Jensen's inequality. This tool provides convexity-type estimates that allow us to prove convergence of nonlinear consensus dynamics in the Wasserstein space of probability measures on $\mathbb{R}^d$.