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, which are 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 $x^\ast$.

For a finite number of particles, a block-wise analysis yields explicit error bounds: individual particles achieve long-time consistency near $x^\ast$, and the global best particle converges to $x^\ast$.

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.

Authors: Bianchi P., Dragomir R.-A., Priser V.
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