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Metropolis-adjusted interacting particle sampling (2312.13889v1)

Published 21 Dec 2023 in stat.CO, cs.NA, and math.NA

Abstract: In recent years, various interacting particle samplers have been developed to sample from complex target distributions, such as those found in Bayesian inverse problems. These samplers are motivated by the mean-field limit perspective and implemented as ensembles of particles that move in the product state space according to coupled stochastic differential equations. The ensemble approximation and numerical time stepping used to simulate these systems can introduce bias and affect the invariance of the particle system with respect to the target distribution. To correct for this, we investigate the use of a Metropolization step, similar to the Metropolis-adjusted Langevin algorithm. We examine Metropolization of either the whole ensemble or smaller subsets of the ensemble, and prove basic convergence of the resulting ensemble Markov chain to the target distribution. Our numerical results demonstrate the benefits of this correction in numerical examples for popular interacting particle samplers such as ALDI, CBS, and stochastic SVGD.

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References (43)
  1. J. Besag. Discussion of “Representations of knowledge in complex systems”. J. Roy. Statist. Soc. Ser. B, 56(4):591–592, 1994.
  2. Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC, New York, NY, 2011.
  3. Consensus-based sampling. Studies in Applied Mathematics, 148(3):1069–1140, 2022.
  4. Neural ordinary differential equations. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
  5. J. A. Christen and C. Fox. A general purpose sampling algorithm for continuous distributions (the t-walk). Bayesian Analysis, 5(2):263 – 281, 2010.
  6. MCMC methods for functions: Modifying old algorithms to make them faster. Statistial Science, 28(3):283–464, 2013.
  7. J. Coullon and R. J. Webber. Ensemble sampler for infinite-dimensional inverse problems. Statistics and Computing, 31:28, 2021.
  8. Approximation and sampling of multivariate probability distributions in the tensor train decomposition. Stat. Comput., 30(3):603–625, 2020.
  9. On the geometry of Stein variational gradient descent. ArXiv, abs/1912.00894, 2019.
  10. M. M. Dunlop and G. Stadler. A gradient-free subspace-adjusting ensemble sampler for infinite-dimensional Bayesian inverse problems. ArXiv, abs/2202.11088, 2022.
  11. V. Gallego and D. R. Insua. Stochastic gradient MCMC with repulsive forces. ArXiv, abs/1812.00071, 2018.
  12. Interacting Langevin diffusions: gradient structure and ensemble Kalman sampler. SIAM Journal on Applied Dynamical Systems, 19(1):412–441, 2020.
  13. Affine invariant interacting Langevin dynamics for Bayesian inference. SIAM Journal on Applied Dynamical Systems, 19(3):1633–1658, 2020.
  14. J. Goodman and J. Weare. Ensemble samplers with affine invariance. Comm. App. Math. and Comp. Sci., (1), 2010.
  15. U. Grenander and M. I. Miller. Representations of knowledge in complex systems. J. Roy. Statist. Soc. Ser. B, 56(4):549–603, 1994.
  16. W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97–109, 1970.
  17. Efficient derivative-free Bayesian inference for large-scale inverse problems. Inverse Problems, 38(12):125006, oct 2022.
  18. Ensemble Kalman methods for inverse problems. Inverse Problems, 29(4):045001, mar 2013.
  19. Sum-of-squares polynomial flow. ICML, 2019.
  20. The variational formulation of the Fokker–Planck equation. SIAM Journal on Mathematical Analysis, 29:1–17, 1998.
  21. A non-asymptotic analysis for Stein variational gradient descent. In Advances in Neural Information Processing Systems, volume 33, pages 4672–4682. Curran Associates, Inc., 2020.
  22. Ensemble preconditioning for Markov chain Monte Carlo simulation. Statistics and Computing, 28(2):277–290, 2018.
  23. Q. Liu. Stein variational gradient descent as gradient flow. In Advances in Neural Information Processing Systems 30, pages 3115–3123. Curran Associates, Inc., 2017.
  24. Q. Liu and D. Wang. Stein variational gradient descent: A general purpose Bayesian inference algorithm. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, page 2378–2386, Red Hook, NY, USA, 2016. Curran Associates Inc.
  25. P. A. Markowich and C. Villani. On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis. volume 19, pages 1–29. 2000. VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999).
  26. Sampling via measure transport: an introduction. In Handbook of uncertainty quantification. Vol. 1, 2, 3, pages 785–825. Springer, Cham, 2017.
  27. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys., 21(6):1087–1092, June 1953.
  28. N. Nüsken and S. Reich. Note on interacting Langevin diffusion: Gradient structure and ensemble Kalman sampler. Technical Report arXiv:1908.10890v1, University of Potsdam, 2019.
  29. N. Nüsken and D. R. M. Renger. Stein variational gradient descent: Many-particle and long-time asymptotics. Foundations of Data Science, 2023.
  30. Mckean–Vlasov SDEs in nonlinear filtering. SIAM Journal on Control and Optimization, 59(6):4188–4215, 2021.
  31. G. Pavliotis. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Texts in Applied Mathematics. Springer New York, 2014.
  32. A consensus-based model for global optimization and its mean-field limit. Mathematical Models & Methods in Applied Sciences, 27(1):183–204, 2017.
  33. S. Reich and S. Weissmann. Fokker–Planck particle systems for Bayesian inference: Computational approaches. SIAM/ASA Journal on Uncertainty Quantification, 9(2):446–482, 2021.
  34. D. Rezende and S. Mohamed. Variational inference with normalizing flows. In F. Bach and D. Blei, editors, Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 1530–1538, Lille, France, 07–09 Jul 2015. PMLR.
  35. C. P. Robert and G. Casella. Monte Carlo Statistical Methods. Texts in Statistics. Springer New York, 2004.
  36. Optimal scaling for various Metropolis–Hastings algorithms. Statistical Science, 16(4):351–367, 2001.
  37. General state space Markov chains and MCMC algorithms. Probability Surveys, 1:20–71, 2004.
  38. Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. Annals of Applied Probability, 16(4):2123–2139, 2006.
  39. Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, 2(4):341–363, 1996.
  40. D. Rudolf and B. Sprungk. On a generalization of the preconditioned Crank–Nicolson Metropolis algorithm. Found. Comput. Math., 18:309–343, 2018.
  41. D. Rudolf and B. Sprungk. Robust random walk-like Metropolis–Hastings algorithms for concentrating posteriors. arXiv:2202.12127, 2022.
  42. S. Vempala and A. Wibisono. Rapid convergence of the unadjusted Langevin algorithm: Isoperimetry suffices. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
  43. S. Weissmann. Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models. Inverse Problems, 38(10):105011, sep 2022.
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