Swarm-based gradient descent meets simulated annealing

We introduce a novel method for non-convex optimization, called Swarm-based Simulated Annealing (SSA), which is at the interface between the swarm-based gradient-descent (SBGD) [J. Lu et. al., ArXiv:2211.17157; E.Tadmor and A. Zenginoglu, Acta Applicandae Math., 190, 2024] and Simulated Annealing (SA) [V. Cerny, J. optimization theory and appl., 45:41-51, 1985; S.Kirkpatrick et. al., Science, 220(4598):671-680, 1983; S. Geman and C.-R. Hwang, SIAM J. Control and Optimization, 24(5):1031-1043, 1986]. Similar to SBGD, we introduce a swarm of agents, each identified with a position, $\mathbf{x}$ and mass $m$, to explore the ambient space. Similar to SA, the agents proceed in the gradient descent direction, and are subject to Brownian motion. The annealing rate, however, is dictated by a decreasing function of their mass. As a consequence, instead of the SA protocol for time-decreasing temperature, we let the swarm decide how to cool down' agents, depending on their accumulated mass over time. The dynamics of masses is coupled with the dynamics of positions: agents at higher ground transfer (part of) their mass to those at lower ground. Consequently, resulting SSA optimizer is dynamically divided between heavier, cooler agents viewed as leaders’ and lighter, warmer agents viewed as `explorers’. Mean-field convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based method as a multi-dimensional global optimizer.