TY  - JOUR
AU  - Bungert, Leon
AU  - Roith, Tim
AU  - Wacker, Philipp
TI  - Polarized consensus-based dynamics for optimization and sampling
JO  - Mathematical programming
VL  - 21
SN  - 0025-5610
CY  - Heidelberg
PB  - Springer
M1  - PUBDB-2024-01832
SP  - 125 - 155
PY  - 2024
N1  - L:MB
AB  - 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.
LB  - PUB:(DE-HGF)16
UR  - <Go to ISI:>//WOS:001236090900001
DO  - DOI:10.1007/s10107-024-02095-y
UR  - https://bib-pubdb1.desy.de/record/607321
ER  -