Polarized consensus-based dynamics for optimization and sampling

• DOI: 10.1007/s10107-024-02095-y
• Publication Date: 2024-05-31T06:01:53Z
• Authors: Leon Bungert; Tim Roith; Philipp Wacker
• Keywords: polarization; sampling; global optimization; 90C26, 35Q93, 35B40, 65N21; Numerical Analysis (math.NA); 01 natural sciences; consensus-based optimization; 510; 49 - Mathematical sciences::4905 - Statistics::490503 - Computational statistics; Optimization and Control (math.OC); FOS: Mathematics; Mathematics - Numerical Analysis; 49 - Mathematical sciences::4903 - Numerical and computational mathematics::490304 - Optimisation; 0101 mathematics; Mathematics - Optimization and Control

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.