Mean-Square Convergence of Particle Swarm Optimization via Stochastic Momentum Analysis

We analyze the standard multi-particle particle swarm optimization (PSO) algorithm with global-best (all-to-all) topology and constant hyperparameters on smooth strongly convex objectives. By rewriting the PSO velocity recursion as a stochastic heavy-ball method acting on a time-varying quadratic surrogate defined by the personal and global bests, and by applying a Lyapunov drift argument in the style of stochastic momentum analyses, we obtain mean-square convergence of particle positions to the unique minimizer and convergence of the best-so-far objective gaps. The deterministic PSO obtained by fixing the random coefficients at their mean values appears as a noise-free special case of the same Lyapunov framework.