2024/05/31 by Leon Bungert, Tim Roith, Philipp Wacker · 10 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Computer science #Distributed Control Multi-Agent Systems #Dynamics (music) #Mathematical optimization #Mathematics #Opinion Dynamics and Social Influence #Sampling (signal processing)
paper · pdf · doi:10.1007/s10107-024-02095-y
published in Mathematical Programming 211(1-2), 125-155 (Springer Science+Business Media)
openalex publication_date 2024/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.