Fast and robust consensus-based optimization via optimal feedback control

TL;DR

This paper introduces controlled-CBO, a feedback-enhanced consensus optimization method that improves convergence to global minima in non-convex problems by using a Hamilton-Jacobi-Bellman-based control law, showing superior performance especially with limited particles.

Contribution

The paper presents a novel controlled-CBO algorithm that integrates a feedback control law derived from HJB equations, enhancing convergence without requiring derivatives of the objective.

Findings

Significantly improved convergence over standard CBO in numerical tests.

Effective with fewer particles and poorly initialized ensembles.

Maintains theoretical tractability in mean-field analysis.

Abstract

We propose a variant of consensus-based optimization (CBO) algorithms, controlled-CBO, which introduces a feedback control term to improve convergence towards global minimizers of non-convex functions in multiple dimensions. The feedback law is a gradient of a numerical approximation to the Hamilton-Jacobi-Bellman (HJB) equation, which serves as a proxy of the original objective function. Thus, the associated control signal furnishes gradient-like information to facilitate the identification of the global minimum without requiring derivative computation from the objective function itself. The proposed method exhibits significantly improved performance over standard CBO methods in numerical experiments, particularly in scenarios involving a limited number of particles, or where the initial particle ensemble is not well positioned with respect to the global minimum. At the same time, the modification keeps the algorithm amenable to theoretical analysis in the mean-field sense. The superior convergence rates are assessed experimentally.