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

Abstract

We study second-order Consensus-Based Optimization (CBO), a derivative-free global optimization algorithm in which the consensus force and the multiplicative exploratory noise act on particle velocities. We prove quantitative uniform-in-time propagation of chaos for the unmodified second-order CBO dynamics, together with an almost uniform-in-time stability estimate for the microscopic particle system. The proof is not a direct adaptation of the first-order CBO argument. Although both first- and second-order CBO have multiplicative noise that degenerates near consensus and a shift-invariant weighted interaction, the kinetic model has an additional structural obstruction: the consensus mechanism and the stochastic forcing act only on the velocity variable, while the position variable evolves by transport. Thus spatial concentration has to be recovered indirectly through velocity dissipation. Moreover, the shift-invariant interaction leaves a translation mode that is not directly damped by the consensus force, so a standard synchronous coupling in the Euclidean phase-space distance does not close uniformly in time. The main idea of the paper is to introduce shifted internal variables that separate the contracting fluctuation modes from the undamped translation mode. In these variables we build a Lyapunov functional with a position-velocity cross term and prove exponential decay of centered moments. This decay is the mechanism that makes the time-dependent coupling coefficient integrable. Combining it with uniform-in-time raw moment bounds, concentration inequalities, stability estimates for the weighted mean, and a Monte Carlo estimate, we obtain the classical Monte Carlo rate for propagation of chaos uniformly in time. The system-to-system stability estimate avoids the sampling error and yields the faster rate \(O(J^{-q})\).