Uniform-in-time propagation of chaos for Consensus-Based Optimization

TL;DR

This paper proves uniform-in-time propagation of chaos and exponential convergence for the Consensus-Based Optimization algorithm, providing new stability estimates and concentration inequalities that improve understanding of its long-term behavior.

Contribution

We establish uniform-in-time propagation of chaos and exponential convergence for CBO, introducing novel stability estimates and concentration inequalities, especially for anisotropic noise cases.

Findings

Propagation of chaos at classical Monte Carlo rate

Exponential convergence near the global minimizer

Prefactor independence from problem dimension in anisotropic noise case

Abstract

We study the derivative-free global optimization algorithm Consensus-Based Optimization (CBO), establishing uniform-in-time propagation of chaos as well as an almost uniform-in-time stability result for the microscopic particle system. Moreover, we prove almost sure exponential convergence of the microscopic CBO system around a point close to the global minimizer. The proof of these results is based on a novel stability estimate for the weighted mean and on a quantitative concentration inequality for the microscopic particle system around the empirical mean. Our propagation of chaos result recovers the classical Monte Carlo rate, with a prefactor that depends explicitly on the parameters of the problem. Notably, in the case of CBO with anisotropic noise, this prefactor is independent of the problem dimension.