Polarized consensus-based dynamics for optimization and sampling

TL;DR

This paper introduces polarized consensus-based dynamics that enhance optimization and sampling methods by allowing detection of multiple minima or modes, using localizing kernels and clustering techniques, with proven convergence properties.

Contribution

The paper develops a novel polarized consensus-based approach that extends existing methods to handle multi-modal functions and distributions, with theoretical convergence guarantees.

Findings

Unbiased in the mean-field regime for Gaussian targets

Converges to the minimizer in the zero temperature limit for convex objectives

Improves efficiency with a clustering-based generalization in high dimensions

Abstract

In this paper we propose polarized consensus-based dynamics in order to make consensus-based optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions with many modes, respectively. For this, we ``polarize'' the dynamics with a localizing kernel and the resulting model can be viewed as a bounded confidence model for opinion formation in the presence of common objective. Instead of being attracted to a common weighted mean as in the original consensus-based methods, which prevents the detection of more than one minimum or mode, in our method every particle is attracted to a weighted mean which gives more weight to nearby particles. We prove that in the mean-field regime the polarized CBS dynamics are unbiased for Gaussian targets. We also prove that in the zero temperature limit and for sufficiently well-behaved strongly convex objectives the solution of the Fokker--Planck equation converges in the Wasserstein-2 distance to a Dirac measure at the minimizer. Finally, we propose a computationally more efficient generalization which works with a predefined number of clusters and improves upon our polarized baseline method for high-dimensional optimization.