Uniform-in-time weak propagation of chaos for consensus-based optimization
Erhan Bayraktar, Ibrahim Ekren, Hongyi Zhou
TL;DR
This work studies uniform-in-time weak propagation of chaos for the consensus-based optimization (CBO) method on a bounded domain. It employs the Delarue–Tse framework, decomposing the weak error through the master equation and analyzing linearized Fokker-Planck equations to obtain exponential decay of relevant derivatives. The main contribution is an $O(N^{-1})$ bound on the weak error, uniformly in time, implying joint convergence of the particle empirical measure to a Dirac mass at the global minimizer in Wasserstein-type metrics as both population size $N$ and time grow. This yields a practical guideline: one can choose $N$ independently of running time to achieve a prescribed tolerance, while the cutoff ensures bounded-domain dynamics essential for the estimates. The results advance theoretical understanding of long-time, gradient-free optimization via interacting particle systems and provide quantitative, time-uniform error control for CBO.
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.