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.

翻译：本文提出极化共识动态，旨在使基于共识的优化（CBO）与采样（CBS）方法适用于具有多个全局最小值的目标函数或多模态分布。为此，我们通过局部化核“极化”动态过程，所得模型可视为存在共同目标时的有界置信观点形成模型。与原始基于共识的方法中所有粒子受吸引至共同加权均值（这阻碍了多个最小值或模态的检测）不同，我们的方法中每个粒子受吸引至一个赋予邻近粒子更高权重的加权均值。我们证明，在平均场框架下，极化CBS动态对于高斯目标是无偏的。我们还证明，在零温极限下，对于足够正则且强凸的目标函数，Fokker-Planck方程的解在Wasserstein-2距离下收敛至最小化处的狄拉克测度。最后，我们提出一种计算效率更高的泛化方法，该方法基于预设数量的聚类，并在高维优化中优于我们的极化基线方法。