Proximal Markov chain Monte Carlo algorithms

By Marcelo Pereyra. 2013

201 citations recorded by the source corpus.

This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that exploits log-concavity to construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau–Yoshida regularisations for convex functions and uses proximity mappings instead of gradient mappings to approximate the continuous-time process. The proposed method complements existing MALA methods in two ways. First, the method is shown to have very robust stability properties and to converge geometrically for many target densities for which other MALA are not geometric, or only if the step size is sufficiently small. Second, the method can be applied to high-dimensional target densities that are not continuously differentiable, a class of distributions that is increasingly used in image processing and machine learning and that is beyond the scope of existing MALA and HMC algorithms. To use this method it is necessary to compute or to approximate efficiently the proximity mappings of the logarithm of the target density. For several popular models, including many Bayesian models used in modern signal and image processing and machine learning, this can be achieved with convex optimisation algorithms and with approximations based on proximal splitting techniques, which can be implemented in parallel. The proposed method is demonstrated on two challenging high-dimensional and non-differentiable models related to image resolution enhancement and low-rank matrix estimation that are not well addressed by existing MCMC methodology.

Subfields: Computational Mathematics, Computational Mechanics, Statistics and Probability

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “ail the non-IID setting, where samples arise from a Markov chain, for example. (iv) Extending the methodology to non-dierentiable models using suitable alternatives to the classical derivative (e.g. Pereyra, 2013). (v) We sketched in Appendix B how orthogonal basis expansions can be used to estimate”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, methodology): “Another promising research area is to use mathematical and algorithmic tools from convex optimisation to design more efﬁcient high-dimensional MCMC algorithms (Pereyra 2015).”
Log-concave sampling: Metropolis-Hastings algorithms are fast! (2018, methodology): “Algorithms related to the Langevin diﬀusion include the Metropolis adjusted Langevin Algorithm (MALA) (Roberts and Tweedie, 1996a; Roberts and Stramer, 2002; Bou-Rabee and Hairer, 2012), the unadjusted Langevin algorithm (ULA) (Parisi, 1981; Grenander and Miller, 1994; Roberts and Tweedie, 1996a; Da”
Approximating Bayesian confidence regions in convex inverse problems (2016, unknown): “The approximations can be computed very efficiently by using standard convex optimisation techniques [3], and are available as a free by-product in any problem that is solved by maximum-a-posteriori (MAP) estimation.”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “…optimal scaling and choice of the discretisation step h , adaptive versions (both discussed in Section 2.4), combinations with proximal operators Pereyra (2013); Schreck et al. (2013), and applications and algorithm development for the inﬁnite-dimensional context Pillai et al. (2012); Cotter et…”
Theoretical guarantees for approximate sampling from smooth and log‐concave densities (2014, background): “They mainly focused on proposing new algorithms of approximate sampling (Girolami and Calderhead, 2011; Pereyra, 2014; Schreck et al., 2013) inspired by the ideas coming from convex optimisation.”