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We investigate the implementation of a new stochastic Kuramoto-Vicsek-typemodel for global optimization of nonconvex functions on the sphere. This modelbelongs to the class of Consensus-Based Optimization. In fact, particles moveon the sphere driven by a drift towards an instantaneous consensus point, whichis computed as a convex combination of particle locations, weighted by the costfunction according to Laplace's principle, and it represents an approximationto a global minimizer. The dynamics is further perturbed by a random vectorfield to favor exploration, whose variance is a function of the distance of theparticles to the consensus point. In particular, as soon as the consensus isreached the stochastic component vanishes. The main results of this paper areabout the proof of convergence of the numerical scheme to global minimizersprovided conditions of well-preparation of the initial datum. The proofcombines previous results of mean-field limit with a novel asymptotic analysis,and classical convergence results of numerical methods for SDE. We presentseveral numerical experiments, which show that the algorithm proposed in thepresent paper scales well with the dimension and is extremely versatile. Toquantify the performances of the new approach, we show that the algorithm isable to perform essentially as good as ad hoc state of the art methods inchallenging problems in signal processing and machine learning, namely thephase retrieval problem and the robust subspace detection.

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