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001     607321
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024 7 _ |a arXiv:2211.05238
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024 7 _ |a 10.1007/s10107-024-02095-y
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100 1 _ |a Bungert, Leon
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245 _ _ |a Polarized consensus-based dynamics for optimization and sampling
260 _ _ |a Heidelberg
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520 _ _ |a 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.
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700 1 _ |a Roith, Tim
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700 1 _ |a Wacker, Philipp
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773 _ _ |a 10.1007/s10107-024-02095-y
|g Vol. 211, no. 1-2, p. 125 - 155
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|t Mathematical programming
|v 21
|y 2024
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