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Controlled Swarm Gradient Dynamics

Louison Aubert

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.

Controlled Swarm Gradient Dynamics

Abstract

We consider the global optimization of a non-convex potential 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 . 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.
Paper Structure (27 sections, 11 theorems, 194 equations, 6 figures, 1 algorithm)

This paper contains 27 sections, 11 theorems, 194 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Acknowledgements.
  2. Objective of the present work.
  3. Swarm gradient dynamics.
  4. Controlled stochastic annealing.
  5. Related Work
  6. Organization of the paper
  7. Mathematical Background
  8. Notation
  9. Optimal transport
  10. Absolutely continuous curves in Wasserstein space
  11. Sobolev spaces
  12. From the Homogeneous Swarm Gradient Dynamics to the Controlled Annealing Strategy
  13. Weak convergence of $\rho_t$
  14. Absolute Continuity of the Curve (rho_t)_t≥0
  15. Motivations and Strategy
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $I \subset \mathbb{R}$ be an open interval, and let $t \mapsto \mu_t \in \mathcal{P}_2(\mathbb{R}^d)$ be an absolutely continuous curve. Then there exists a time-dependent vector field $v_t(x) : I \times \mathbb{R}^d \to \mathbb{R}^d$ such that the pair $(\mu_t, v_t)$ satisfies the continuity eq Moreover, it holds that $\|v_t\|_{L^2(\mu_t, \mathbb{R}^d)} \in L^1(I)$. Conversely, if such a fiel

Figures (6)

  • Figure 1: 1D double-well potential $U(x)$.
  • Figure 2: Particle heatmaps for CSA (top-left) and Controlled Swarm Gradient (top-right for $m=2$, bottom for $m=6$). All runs use 100 experiments with 100 particles each. Time step $\Delta t = 0.002$, velocity estimated every 20 iterations ($h = 0.04$), except for the bottom-right panel where $h = 0.02$.
  • Figure 3: Evolution of the median of the quantity $\min_{1\le i\le 5} \min_t X_t^{(i)}$ for CSA and CSG with parameters $m=2$ and $m=6$. Each curve represents the median over 1000 experiments using a quadratic cooling schedule : $\beta(t)=0.25 + 25\,t^{2}$.
  • Figure 4: 2D Six-Hump Camel Function.
  • Figure 5: Median of $\min_{1\le i\le 5} U(X_t^i)$ over $1000$ experiments, for CSA and CSG ($m=2$ and $m=6$). Parameters: $\Delta t = 0.002$, velocity refresh rate $h=0.04$, linear cooling schedule $\beta(t)=0.25+25t$.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1.1: Absolutely continuous curves and metric derivative
  • Theorem 1.2: Absolutely continuous curves and the continuity equation ambrosio2008gradient
  • Proposition 1.3: Transport map characterization of the velocity field ambrosio2008gradient
  • Theorem 1.4: Characterization of the minimal velocity field ambrosio2008gradient
  • Definition 1.5: $L^2$-Poincaré constant
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Lemma 3.2
  • ...and 13 more