Polarized consensus-based dynamics for optimization and sampling

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. The resulting dynamics possess mean-field interpretations with Fokker--Planck equations that are structurally similar to the ones of original CBO and CBS, and we prove that the polarized CBS dynamics is unbiased in case of a Gaussian target. We also 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.