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

Nicolai Gerber, Franca Hoffmann, Dohyeon Kim, Urbain Vaes

TL;DR

This work addresses the rigorous connection between the finite-particle CBO dynamics and its mean-field limit in the long-time regime. By developing a stable, synchronous coupling framework and new concentration tools for the weighted mean, the authors prove uniform-in-time propagation of chaos for the original CBO algorithm and establish almost UiT stability between particle systems driven by the same noise. The results yield explicit, parameter-dependent constants and show that anisotropic noise can render the Monte Carlo prefactor independent of dimension, a notable property for high-dimensional optimization. The findings bridge microscopic particle behavior with macroscopic mean-field descriptions, enhancing theoretical guarantees for gradient-free global optimization via CBO and informing practical parameter choices. Overall, the paper provides a solid UiT probabilistic analysis of CBO, complementing existing PDE-level mean-field results and offering quantitative, implementable error bounds.

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. 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.

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

TL;DR

This work addresses the rigorous connection between the finite-particle CBO dynamics and its mean-field limit in the long-time regime. By developing a stable, synchronous coupling framework and new concentration tools for the weighted mean, the authors prove uniform-in-time propagation of chaos for the original CBO algorithm and establish almost UiT stability between particle systems driven by the same noise. The results yield explicit, parameter-dependent constants and show that anisotropic noise can render the Monte Carlo prefactor independent of dimension, a notable property for high-dimensional optimization. The findings bridge microscopic particle behavior with macroscopic mean-field descriptions, enhancing theoretical guarantees for gradient-free global optimization via CBO and informing practical parameter choices. Overall, the paper provides a solid UiT probabilistic analysis of CBO, complementing existing PDE-level mean-field results and offering quantitative, implementable error bounds.

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. 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.
Paper Structure (40 sections, 13 theorems, 161 equations, 2 tables)

This paper contains 40 sections, 13 theorems, 161 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Mathematical setting
  4. Propagation of chaos
  5. Our contributions
  6. Plan of the paper
  7. Notation
  8. Main results
  9. Uniform-in-time propagation of chaos
  10. Almost uniform-in-time stability for the interacting particle system
  11. Comments on the proof strategy
  12. Proof of the main results
  13. Proof of Uniform-in-time propagation of chaos
  14. Bounding $\mathcal{A}_1(t)$
  15. Bounding $\mathcal{A}_2(t)$
  16. ...and 25 more sections

Key Result

Theorem 2.1

Fix a probability measure ${ \macc@depth1 \frozen@everymath{\mathgroup\macc@group} \macc@set@skewchar \macc@nested@a111{} } _0\in \mathcal{P}(\mathbf R^d)$ with finite moments of all orders. Let $(\Omega, \mathcal{F}, \mathbf P)$ be a probability space supporting initial i.i.d. positions $\bigl( X^ To this system we couple the system of i.i.d. mean-field particles starting at the same initial po

Theorems & Definitions (32)

  • Theorem 2.1: Uniform-in-time mean-field limit
  • Remark 2.2
  • Theorem 2.3
  • proof
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2: Exponential decay of centered moments
  • ...and 22 more