Uniform-in-time weak propagation of chaos for consensus-based optimization

TL;DR

This paper proves that the consensus-based optimization (CBO) method's particle system converges uniformly over time to the global minimum, with error decreasing as the inverse of the number of particles, using advanced probabilistic techniques.

Contribution

It establishes the first uniform-in-time weak propagation of chaos result for CBO, demonstrating convergence in Wasserstein metrics and providing a rigorous theoretical foundation.

Findings

Weak error order is O(N^{-1}) uniformly in time.

Empirical distribution converges to the global minimizer.

Results hold for bounded search domains.

Abstract

We study the uniform-in-time weak propagation of chaos for the consensus-based optimization (CBO) method on a bounded searching domain. We apply the methodology for studying long-time behaviors of interacting particle systems developed in the work of Delarue and Tse (ArXiv:2104.14973). Our work shows that the weak error has order $O(N^{-1})$ uniformly in time, where $N$ denotes the number of particles. The main strategy behind the proofs are the decomposition of the weak errors using the linearized Fokker-Planck equations and the exponential decay of their Sobolev norms. Consequently, our result leads to the joint convergence of the empirical distribution of the CBO particle system to the Dirac-delta distribution at the global minimizer in population size and running time in Wasserstein-type metrics.