%0 Journal Article
%A Bungert, Leon
%A Roith, Tim
%A Wacker, Philipp
%T Polarized consensus-based dynamics for optimization and sampling
%J Mathematical programming
%V 21
%@ 0025-5610
%C Heidelberg
%I Springer
%M PUBDB-2024-01832
%P 125 - 155
%D 2024
%Z L:MB
%X 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.
%F PUB:(DE-HGF)16
%9 Journal Article
%U <Go to ISI:>//WOS:001236090900001
%R 10.1007/s10107-024-02095-y
%U https://bib-pubdb1.desy.de/record/607321