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It is well known that Markov chain Monte Carlo (MCMC) methods scale poorlywith dataset size. A popular class of methods for solving this issue isstochastic gradient MCMC. These methods use a noisy estimate of the gradient ofthe log posterior, which reduces the per iteration computational cost of thealgorithm. Despite this, there are a number of results suggesting thatstochastic gradient Langevin dynamics (SGLD), probably the most popular ofthese methods, still has computational cost proportional to the dataset size.We suggest an alternative log posterior gradient estimate for stochasticgradient MCMC, which uses control variates to reduce the variance. We analyseSGLD using this gradient estimate, and show that, under log-concavityassumptions on the target distribution, the computational cost required for agiven level of accuracy is independent of the dataset size. Next we show that adifferent control variate technique, known as zero variance control variatescan be applied to SGMCMC algorithms for free. This post-processing stepimproves the inference of the algorithm by reducing the variance of the MCMCoutput. Zero variance control variates rely on the gradient of the logposterior; we explore how the variance reduction is affected by replacing thiswith the noisy gradient estimate calculated by SGMCMC.

Control Variates for Stochastic Gradient MCMC