  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.
