Polarized consensus-based dynamics for optimization and sampling
| dc.contributor.author | Wacker, Philipp | |
| dc.contributor.author | Bungert , Leon | |
| dc.contributor.author | Roith , Tim | |
| dc.date.accessioned | 2024-06-14T01:53:06Z | |
| dc.date.available | 2024-06-14T01:53:06Z | |
| dc.date.issued | 2024 | |
| dc.description.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. | |
| dc.identifier.citation | Wacker P, Bungert L, Roith T (2024). Polarized consensus-based dynamics for optimization and sampling. Mathematical Programming. | |
| dc.identifier.doi | http://doi.org/10.1007/s10107-024-02095-y | |
| dc.identifier.issn | 0025-5610 | |
| dc.identifier.uri | https://hdl.handle.net/10092/107141 | |
| dc.rights | OpenAccess This article is licensed under a CreativeCommonsAttribution 4.0 InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. | |
| dc.rights.uri | http://hdl.handle.net/10092/17651 | |
| dc.subject | global optimization | |
| dc.subject | consensus-based optimization | |
| dc.subject | polarization | |
| dc.subject | sampling | |
| dc.subject.anzsrc | 49 - Mathematical sciences::4903 - Numerical and computational mathematics::490304 - Optimisation | |
| dc.subject.anzsrc | 49 - Mathematical sciences::4905 - Statistics::490503 - Computational statistics | |
| dc.title | Polarized consensus-based dynamics for optimization and sampling | |
| dc.type | Journal Article | |
| uc.college | Faculty of Engineering | |
| uc.department | Mathematics and Statistics |