From Consensus-Based Optimization to Particle Swarm Optimization: Convergence Guarantees under Drift-Diffusion Coupling

Particle swarm optimization (PSO) is a widely used algorithm featured in many state-of-the-art optimization toolkits, but rigorous performance guarantees are still lacking, in part because the classical dynamics do not admit a natural mean-field description. Consensus-based optimization (CBO) with memory can be obtained from a regularized version of PSO and does admit such a description, giving access to rigorous convergence theory. However, this theory relies on being able to choose the drift and diffusion strengths independently, whereas PSO couples them through a single coefficient. We study how this coupling affects existing convergence guarantees for CBO and its memory variants. We show, by an explicit construction, that the PSO coupling still permits a non-empty set of admissible parameters. Nonetheless, the parameter limits that recover the PSO model degenerate this set, so the coupling is a genuine obstruction to applying existing convergence results to PSO. We illustrate the resulting tradeoffs numerically.