Non-stationary phase of the MALA algorithm
- Creators
- Kuntz, Juan
- Ottobre, Michela
- Stuart, Andrew M.
Abstract
The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, π^N, with Lebesgue density on R^N; it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm as a function of N. The measure of efficiency that we consider in this paper is the expected squared jumping distance (ESJD), introduced in Roberts et al. (Ann Appl Probab 7(1):110–120, 1997). To determine how the cost of the algorithm (in terms of ESJD) increases with dimension N, we adopt the widely used approach of deriving a diffusion limit for the Markov chain produced by the MALA algorithm. We study this problem for a class of target measures which is not in product form and we address the situation of practical relevance in which the algorithm is started out of stationarity. We thereby significantly extend previous works which consider either measures of product form, when the Markov chain is started out of stationarity, or non-product measures (defined via a density with respect to a Gaussian), when the Markov chain is started in stationarity. In order to work in this non-stationary and non-product setting, significant new analysis is required. In particular, our diffusion limit comprises a stochastic PDE coupled to a scalar ordinary differential equation which gives a measure of how far from stationarity the process is. The family of non-product target measures that we consider in this paper are found from discretization of a measure on an infinite dimensional Hilbert space; the discretised measure is defined by its density with respect to a Gaussian random field. The results of this paper demonstrate that, in the non-stationary regime, the cost of the algorithm is of O(N^(1/2)) in contrast to the stationary regime, where it is of O(N^(1/3)).
Additional Information
© 2018 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 3 August 2017; First Online: 17 April 2018. A.M. Stuart acknowledges support from AMS, DARPA, EPSRC, ONR. J. Kuntz gratefully acknowledges support from the BBSRC in the form of the Ph.D. studentship BB/F017510/1. M. Ottobre and J.Kuntz gratefully acknowledge financial support from the Edinburgh Mathematical Society.Attached Files
Published - Kuntz2018_Article_Non-stationaryPhaseOfTheMALAAl.pdf
Submitted - 1608.08379v1.pdf
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Additional details
- PMCID
- PMC6411168
- Eprint ID
- 73037
- Resolver ID
- CaltechAUTHORS:20161220-175620681
- American Mathematical Society
- Defense Advanced Research Projects Agency (DARPA)
- Engineering and Physical Sciences Research Council (EPSRC)
- Office of Naval Research (ONR)
- BB/F017510/1
- Biotechnology and Biological Sciences Research Council (BBSRC)
- Edinburgh Mathematical Society
- Created
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2016-12-21Created from EPrint's datestamp field
- Updated
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2022-03-11Created from EPrint's last_modified field