Mean-field limits for Consensus-Based Optimization and Sampling

TL;DR

This paper establishes quantitative mean-field limit results for consensus-based optimization and sampling algorithms, ensuring finite particle systems approximate their mean-field models with controlled error, and introduces new stability and well-posedness results.

Contribution

It extends classical mean-field analysis by handling non-globally Lipschitz coefficients and provides new stability estimates for weighted moments and covariance.

Findings

Quantitative convergence rates for particle systems to mean-field models.

New well-posedness results for the particle and mean-field systems.

Stability estimates for weighted mean and covariance.

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.