Mean-field limits for Consensus-Based Optimization and Sampling

¹ Hausdorff Center for Mathematics, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany (before September 2023)
² Institute of Applied Analysis, Ulm University, Germany (since October 2024)
³ Department of Computing and Mathematical Sciences, Caltech, USA
⁴ MATHERIALS project-team, Inria Paris
⁵ CERMICS, École des Ponts, France

^* Corresponding author: nicolai.gerber@uni-ulm.de

Received: 25 November 2024
Accepted: 2 July 2025

Abstract

For algorithms based on interacting particle systems that admit a mean-field description, convergence analysis is often more accessible at the mean-field level. In order to transfer convergence results obtained at the mean-field level to the finite ensemble size setting, it is desirable to show that the particle dynamics converge in an appropriate sense to the corresponding mean-field dynamics. In this paper, we prove quantitative mean-field limit results for two related interacting particle systems: Consensus-Based Optimization and Consensus-Based Sampling. Our approach requires a generalization of Sznitman’s classical argument: in order to circumvent issues related to the lack of global Lipschitz continuity of the coefficients, we discard an event of small probability, the contribution of which is controlled using moment estimates for the particle systems. In addition, we present new results on the well-posedness of the particle systems and their mean-field limit, and provide novel stability estimates for the weighted mean and the weighted covariance.