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Consistency and fluctuations for stochastic gradient Langevin dynamics
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Teh, Yee Whye, Thiery, Alexandre H. and Vollmer, Sebastian (2016) Consistency and fluctuations for stochastic gradient Langevin dynamics. Journal of Machine Learning Research, 17 . pp. 193-225. doi:10.5555/2946645.2946652 ISSN 1532-4435.
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Official URL: http://dx.doi.org/10.5555/2946645.2946652
Abstract
Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally expensive. Both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem by generating proposals which are only based on a subset of the data, by skipping the accept-reject step and by using decreasing step-sizes sequence (δm)m≥0.
We provide in this article a rigorous mathematical framework for analysing this algorithm. We prove that, under verifiable assumptions, the algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence (δm)m≥0. We leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm: it is asymptotically optimal to use a step-size sequence of the type δm = m-1/3, leading to an algorithm whose mean squared error (MSE) decreases at rate O(m-1/3).
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Markov processes, Monte Carlo method , Langevin equations , Stochastic differential equations , Big data | ||||||||||||
| Journal or Publication Title: | Journal of Machine Learning Research | ||||||||||||
| Publisher: | M I T Press | ||||||||||||
| ISSN: | 1532-4435 | ||||||||||||
| Official Date: | 1 January 2016 | ||||||||||||
| Dates: |
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| Volume: | 17 | ||||||||||||
| Page Range: | pp. 193-225 | ||||||||||||
| DOI: | 10.5555/2946645.2946652 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
| Date of first compliant deposit: | 30 October 2019 | ||||||||||||
| Date of first compliant Open Access: | 30 October 2019 | ||||||||||||
| RIOXX Funder/Project Grant: |
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