Exponential stability of finite-$N$ consensus-based optimization

TL;DR

This paper investigates the stability of finite-agent Consensus-Based Optimization (CBO), demonstrating that key stability properties like exponential convergence hold for finite populations, with quantitative convergence rates provided.

Contribution

It provides the first detailed stability analysis of finite-agent CBO, extending previous mean-field results to finite populations with explicit convergence estimates.

Findings

Almost sure exponential convergence in finite-agent CBO

Mean square exponential convergence persists for finite N

Quantitative estimates on convergence rates

Abstract

We study the finite-agent behavior of Consensus-Based Optimization (CBO), a recent metaheuristic for the global minimization of a function, that combines drift toward a consensus estimate with stochastic exploration. While previous analyses focus on asymptotic mean-field limits, we investigate the stability properties of CBO for finite population size \( N \). Following a hierarchical approach, we first analyze a deterministic formulation of the algorithm and then extend our results to the fully stochastic setting governed by a system of stochastic differential equations. Our analysis reveals that essential stability properties, including almost sure and mean square exponential convergence, persist in both regimes and provides sharp quantitative estimates on the rates of convergence.