Mean-field limits for Consensus-Based Optimization and Sampling

TL;DR

The paper derives quantitative mean-field limits for Consensus-Based Optimization and Consensus-Based Sampling, providing explicit rates of convergence from finite particle systems to their mean-field descriptions under precise growth and regularity assumptions on the objective and diffusion. It introduces two complementary strategies—indicator sets and stopping times—to handle non-global Lipschitz interactions and to obtain sharp $J^{-1/2}$ rates in both $L^p$ and pathwise Wasserstein metrics, along with new well-posedness and stability results for weighted moments. A key technical advance is the improved Wasserstein stability for the weighted mean and, crucially, for the square root of the weighted covariance, enabling propagation-of-chaos proofs with explicit rates. Numerical experiments with the Ackley function corroborate the theoretical rates and demonstrate practical convergence behavior for moderate to large particle numbers. The work lays a rigorous foundation for transferring mean-field analysis to finite ensembles in non-globally Lipschitz consensus-based dynamics and opens pathways to broader growth regimes and extensions.

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.

Figures (2)

• Figure 1: Interdependence of the results contained in this paper.
• Figure 2: Left: Ackley function. Right: Monte Carlo estimator $\widehat{E}(J)$ of $E(J)$ for $J\in \{10 \cdot 2^k\}_{k=0}^{14}$, and straight line exhibiting the asymptotic scaling as $J^{-1}$ of $E(J)$. Error bars corresponding to two standard deviations are depicted on both sides of the Monte Carlo estimator.

Theorems & Definitions (49)

• Remark 2.1
• Theorem 2.2: Existence and uniqueness for the particle models
• Theorem 2.3: CBO: Existence and uniqueness for the mean-field SDE
• Theorem 2.4: CBS: Existence and uniqueness for the mean-field SDE
• Lemma 2.5: Bound on the probability of large excursions
• proof
• Theorem 2.6: Mean-field limit for CBO
• Remark 2.7
• Remark 2.8
• proof : Proof of \ref{['thm:mfl-cbo']}