Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

Nicolas Brosse, Alain Durmus, Éric Moulines, Marcelo Pereyra
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:319-342, 2017.

Abstract

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $K$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $K$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.

Cite this Paper

BibTeX
@InProceedings{pmlr-v65-brosse17a, title = {Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo}, author = {Brosse, Nicolas and Durmus, Alain and Moulines, Éric and Pereyra, Marcelo}, booktitle = {Proceedings of the 2017 Conference on Learning Theory}, pages = {319--342}, year = {2017}, editor = {Kale, Satyen and Shamir, Ohad}, volume = {65}, series = {Proceedings of Machine Learning Research}, month = {07--10 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v65/brosse17a/brosse17a.pdf}, url = {https://proceedings.mlr.press/v65/brosse17a.html}, abstract = { This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $K$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $K$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.} }
Endnote
%0 Conference Paper %T Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo %A Nicolas Brosse %A Alain Durmus %A Éric Moulines %A Marcelo Pereyra %B Proceedings of the 2017 Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2017 %E Satyen Kale %E Ohad Shamir %F pmlr-v65-brosse17a %I PMLR %P 319--342 %U https://proceedings.mlr.press/v65/brosse17a.html %V 65 %X This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $K$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $K$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.
APA
Brosse, N., Durmus, A., Moulines, É. & Pereyra, M.. (2017). Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:319-342 Available from https://proceedings.mlr.press/v65/brosse17a.html.

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