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

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.

References in corpus (4)

• Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks
• On the mean-field limit for the consensus-based optimization
• Binary interaction methods for high dimensional global optimization and machine learning
• Random batch methods (RBM) for interacting particle systems

Cited by in corpus (7)

• On the mean-field limit for the consensus-based optimization
• Convergence of Anisotropic Consensus-Based Optimization in Mean-Field Law
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• Leveraging Memory Effects and Gradient Information in Consensus-Based Optimization: On Global Convergence in Mean-Field Law
• Consensus based optimization with memory effects: random selection and applications
• Self-interacting CBO: Existence, uniqueness, and long-time convergence
• Consensus-Based Optimization with Truncated Noise