Anisotropic Diffusion in Consensus-based Optimization on the Sphere

TL;DR

This paper introduces an anisotropic stochastic approach to consensus-based optimization on the sphere, proving convergence to global minimizers and demonstrating superior performance in high-dimensional, non-smooth, and non-convex problems.

Contribution

It presents a novel anisotropic stochastic term in consensus-based optimization, enabling dimension-independent parameter scaling and improved exploration.

Findings

Proves convergence of the proposed algorithm to global minimizers.

Shows the method outperforms isotropic noise-based algorithms in experiments.

Demonstrates effectiveness in high-dimensional and complex optimization problems.

Abstract

In this paper we are concerned with the global minimization of a possibly non-smooth and non-convex objective function constrained on the unit hypersphere by means of a multi-agent derivative-free method. The proposed algorithm falls into the class of the recently introduced Consensus-Based Optimization. In fact, agents move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of agent locations, weighted by the cost function according to Laplaces principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by an anisotropic random vector field to favor exploration. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof of convergence combines a mean-field limit result with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. The main innovation with respect to previous work is the introduction of an anisotropic stochastic term, which allows us to ensure the independence of the parameters of the algorithm from the dimension and to scale the method to work in very high dimension. We present several numerical experiments, which show that the algorithm proposed in the present paper is extremely versatile and outperforms previous formulations with isotropic stochastic noise.