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Consensus-based sampling

Carrillo, J. A. and Hoffmann, F. and Stuart, A. M. and Vaes, U. (2022) Consensus-based sampling. Studies in Applied Mathematics, 148 (3). pp. 1069-1140. ISSN 0022-2526. doi:10.1111/sapm.12470.

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We propose a novel method for sampling and optimization tasks based on a stochastic interacting particle system. We explain how this method can be used for the following two goals: (i) generating approximate samples from a given target distribution and (ii) optimizing a given objective function. The approach is derivative-free and affine invariant, and is therefore well-suited for solving inverse problems defined by complex forward models: (i) allows generation of samples from the Bayesian posterior and (ii) allows determination of the maximum a posteriori estimator. We investigate the properties of the proposed family of methods in terms of various parameter choices, both analytically and by means of numerical simulations. The analysis and numerical simulation establish that the method has potential for general purpose optimization tasks over Euclidean space; contraction properties of the algorithm are established under suitable conditions, and computational experiments demonstrate wide basins of attraction for various specific problems. The analysis and experiments also demonstrate the potential for the sampling methodology in regimes in which the target distribution is unimodal and close to Gaussian; indeed we prove that the method recovers a Laplace approximation to the measure in certain parametric regimes and provide numerical evidence that this Laplace approximation attracts a large set of initial conditions in a number of examples.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Carrillo, J. A.0000-0001-8819-4660
Hoffmann, F.0000-0002-1182-5521
Stuart, A. M.0000-0001-9091-7266
Vaes, U.0000-0002-7629-7184
Additional Information:© 2022 The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Issue Online: 02 March 2022; Version of Record online: 05 January 2022; Manuscript accepted: 23 October 2021; Manuscript revised: 18 September 2021; Manuscript received: 24 May 2021. The authors are grateful to Zehua Lai for pointing out that the Poincaré inequality could be employed for proving Lemma 5. JAC was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 883363) and by EPSRC grant number EP/T022132/1. JAC and UV were also supported by EPSRC grant number EP/P031587/1. FH was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. AMS is supported by NSF (award AGS-1835860), by NSF (award DMS-1818977) and by the Office of Naval Research (award N00014-17-1-2079). UV was also supported by the Fondation Sciences Mathématiques de Paris (FSMP), through a postdoctoral fellowship in the “mathematical interactions” program.
Funding AgencyGrant Number
European Research Council (ERC)883363
Engineering and Physical Sciences Research Council (EPSRC)EP/T022132/1
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
Deutsche Forschungsgemeinschaft (DFG)390685813 - GZ 2047/1
Office of Naval Research (ONR)N00014-17-1-2079
Fondation Sciences Mathématiques de Paris (FSMP)UNSPECIFIED
Subject Keywords:optimization; sampling; stochastic interacting particle systems
Issue or Number:3
Record Number:CaltechAUTHORS:20210719-210142693
Persistent URL:
Official Citation:Consensus-based sampling. Carrillo, JA, Hoffmann, F, Stuart, AM, Vaes, U. Consensus-based sampling. Stud Appl Math. 2022; 148: 1069–1140.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109920
Deposited By: George Porter
Deposited On:19 Jul 2021 21:27
Last Modified:23 Mar 2022 22:36

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