Controlled Swarm Gradient Dynamics

Abstract

We consider the global optimization of a non-convex potential $U : \mathbb{R}^d \to \mathbb{R}$ and extend the controlled simulated annealing framework introduced by Molin et al. (2026) to the class of swarm gradient dynamics, a family of Langevin-type mean-field diffusions whose noise intensity depends locally on the marginal density of the process. Building on the time-homogeneous model of Huang and Malik (2025), we first analyze its invariant probability density and show that, as the inverse temperature parameter tends to infinity, it converges weakly to a probability measure supported on the set of global minimizers of $U$. This result justifies using this family of invariant measures as an annealing curve in a controlled swarm setting. Given an arbitrary non-decreasing cooling schedule, we then prove the existence of a velocity field solving the continuity equation associated with the curve of invariant densities. Superimposing this field onto the swarm gradient dynamics yields a well-posed controlled process whose marginal law follows exactly the prescribed annealing curve. As a consequence, the controlled swarm dynamics converges toward global minimizers with, in principle, arbitrarily fast convergence rates, entirely dictated by the choice of the cooling schedule. Finally, we discuss an algorithmic implementation of the controlled dynamics and compare its performance with controlled simulated annealing, highlighting some numerical limitations.