Consensus-Based Optimization Beyond Finite-Time Analysis

We analyze a zeroth-order particle algorithm for the global optimization of a non-convex function, focusing on a variant of Consensus-Based Optimization (CBO) with small but fixed noise intensity. Unlike most previous studies restricted to finite horizons, we investigate its long-time behavior with fixed parameters. In the mean-field limit, a quantitative Laplace principle shows exponential convergence to a neighborhood of the minimizer x * . For finitely many particles, a block-wise analysis yields explicit error bounds: individual particles achieve long-time consistency near x * , and the global best particle converge to x * . The proof technique combines a quantitative Laplace principle with block-wise control of Wasserstein distances, avoiding the exponential blow-up typical of Grönwall-based estimates.