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

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.

💡 Research Summary

This paper investigates the derivative‑free global optimization method known as Consensus‑Based Optimization (CBO). The authors establish two major theoretical results for the continuous‑time particle system underlying CBO: (i) a uniform‑in‑time propagation of chaos, meaning that the empirical measure of a finite‑size particle system stays at a Monte Carlo rate (O(J^{-1})) from the mean‑field limit for all times (t\ge0); and (ii) an almost uniform‑in‑time stability estimate for the particle system, which together imply almost sure exponential convergence of the particles toward a point close to the global minimizer.

The CBO dynamics are given by
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