# Control Variates for Stochastic Gradient MCMC

## TL;DR

This paper introduces control variates to stochastic gradient MCMC methods, significantly reducing variance and computational cost, making the algorithms more scalable for large datasets under certain conditions.

## Contribution

It proposes a novel variance reduction technique for stochastic gradient MCMC using control variates, improving efficiency and scalability of the methods.

## Key findings

- Variance reduction leads to dataset-size independent computational cost.
- Zero variance control variates can be applied for free to improve inference.
- Analysis under log-concavity assumptions shows theoretical efficiency gains.

## 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. 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.