A modified Consensus-Based Optimization model: consensus formation and uniform-in-time propagation of chaos

TL;DR

This paper introduces a modified consensus-based optimization model with a regularized Gibbs weight, providing a unified theoretical framework that guarantees exponential consensus, uniform propagation of chaos, and convergence to the global minimizer under relaxed assumptions.

Contribution

It develops a new stabilized model for consensus-based optimization that simplifies analysis and broadens applicability by removing traditional regularity constraints.

Findings

Particles exponentially converge to a common limit near the global minimizer

Empirical measures converge uniformly to McKean--Vlasov dynamics

Mean-field system achieves deterministic consensus and approaches the global minimizer

Abstract

We introduce a modified Consensus-Based Optimization model that admits a fully unified and rigorous analysis of its finite-particle dynamics, the associated McKean--Vlasov equation, and their optimization behavior under a single set of structural framework. The key ingredient is a regularized Gibbs weight that stabilizes the consensus point and avoids degeneracies present in the classical formulation, eliminating the need for cutoffs, rescaling, or boundedness assumptions on the objective function. Our first main result establishes large-time consensus for the particle system: when the drift exceeds an explicit threshold, all particles converge exponentially to a common random limit that concentrates near the global minimizer. Our second result proves uniform-in-time propagation of chaos, providing quantitative and dimension-free convergence of the empirical measure to the McKean--Vlasov dynamics. Finally, we show that the mean-field system reaches deterministic consensus and that its consensus point approaches the global minimizer in the regime of highly concentrated Gibbs weights. Together, these results yield a unified and internally consistent theoretical framework for consensus-based optimization under substantially relaxed regularity assumptions on the objective function.