Consensus-Based Optimization on the Sphere: Convergence to Global Minimizers and Machine Learning

TL;DR

This paper introduces a stochastic consensus-based optimization method on the sphere, proving its convergence to global minimizers and demonstrating its effectiveness in high-dimensional machine learning tasks like phase retrieval and subspace detection.

Contribution

The paper develops a new stochastic Kuramoto-Vicsek-type model for global optimization on the sphere, with convergence proofs and practical validation in machine learning applications.

Findings

The algorithm converges to global minimizers under certain initial conditions.

It scales effectively with high-dimensional data.

Performs comparably to state-of-the-art methods in signal processing tasks.

Abstract

We investigate the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. 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 combines previous results of mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the algorithm proposed in the present paper scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.