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Control variates for stochastic gradient MCMC

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>1/05/2019
<mark>Journal</mark>Statistics and Computing
Issue number3
Volume29
Number of pages17
Pages (from-to)599-615
Publication statusPublished
Early online date4/08/18
Original languageEnglish

Abstract

It is well known that Markov chain Monte Carlo (MCMC) methods scale poorly with dataset size. A popular class of methods for solving this issue is stochastic gradient MCMC (SGMCMC). These methods use a noisy estimate of the gradient of the log-posterior, which reduces the per iteration computational cost of the algorithm. Despite this, there are a number of results suggesting that stochastic gradient Langevin dynamics (SGLD), probably the most popular of these methods, still has computational cost proportional to the dataset size. We suggest an alternative log-posterior gradient estimate for stochastic gradient MCMC which uses control variates to reduce the variance. We analyse SGLD using this gradient estimate, and show that, under log-concavity assumptions on the target distribution, the computational cost required for a given level of accuracy is independent of the dataset size. Next we show that a different control variate technique, known as zero variance control variates, can be applied to SGMCMC algorithms for free. This post-processing step improves the inference of the algorithm by reducing the variance of the MCMC output. Zero variance control variates rely on the gradient of the log-posterior; we explore how the variance reduction is affected by replacing this with the noisy gradient estimate calculated by SGMCMC.

Bibliographic note

The final publication is available at Springer via https://doi.org/10.1007/s11222-018-9826-2