Simulated annealing finds global minima by gradually lowering temperature: at high temperature the system explores freely, at low temperature it settles. The cooling schedule determines everything -- too fast and the system freezes in a local minimum, too slow and computation is wasted. Louison Aubert extends controlled annealing to swarm gradient dynamics, where the noise intensity depends on the marginal density of the swarm itself. Instead of each particle experiencing a fixed temperature, the temperature at any point reflects how many particles are already there.
This makes the thermostat collective rather than individual. In standard simulated annealing, each particle cools on a schedule that is indifferent to the rest of the population. In swarm dynamics, a region crowded with particles has different effective noise than a region with few. The invariant density converges to a measure supported on global minimizers as inverse temperature increases, and the controlled dynamics can achieve this convergence at arbitrarily fast rates by superimposing a velocity field derived from the target annealing schedule. The practical consequence is that the cooling schedule is no longer an external imposition but an emergent property of the swarm's spatial distribution. A cluster forming around a local minimum changes the local temperature, potentially destabilizing the cluster and pushing particles toward better regions. The swarm optimizes not by following a clock but by reading its own density -- a thermostat that takes its reading from the crowd rather than from the wall.
(arXiv:2603.12039)