Dynamics of measure-valued agents in the space of probabilities

TL;DR

This paper introduces a new multi-agent consensus dynamic where agents are probability measures evolving via Wasserstein metric, with applications to measure-based optimization.

Contribution

It develops a measure differential inclusion framework for agent evolution and demonstrates its use in measure optimization problems.

Findings

Existence of solutions for measure differential inclusions with compact support

Application to measure-based minimization problems

Numerical example with particle approximation

Abstract

Motivated by the development of dynamics in probability spaces, we propose a novel multi-agent dynamic of consensus type where each agent is a probability measure. The agents move instantaneously towards a weighted barycenter of the ensemble according to the 2-Wasserstein metric. We mathematically describe the evolution as a system of measure differential inclusions and show the existence of solutions for compactly supported initial data. Inspired by the consensus-based optimization, we apply the multi-agent system to solve a minimization problem over the space of probability measures. In the small numerical example, each agent is described by a particle approximation and aims to approximate a target measure.