# Control Variates for Stochastic Gradient MCMC

arXiv: https://arxiv.org/abs/1706.05439 · HTML: https://mlsanity.com/paper/1706.05439/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2017.

## Built on by (most cited first, 25 of 25)

- [QLSD: Quantised Langevin stochastic dynamics for Bayesian federated learning](https://mlsanity.com/paper/2106.00797/) (2021)
- [Efficient Attention via Control Variates](https://mlsanity.com/paper/2302.04542/) (2023)
- [Low-Precision Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2206.09909/) (2022)
- [Efficient and Generalizable Tuning Strategies for Stochastic Gradient MCMC](https://mlsanity.com/paper/2105.13059/) (2021)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [SwISS: A Scalable Markov chain Monte Carlo Divide-and-Conquer Strategy](https://mlsanity.com/paper/2208.04080/) (2022)
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- [Tuning Stochastic Gradient Algorithms for Statistical Inference via Large-Sample Asymptotics](https://mlsanity.com/paper/2207.12395/) (2022)
- [Langevin Quasi-Monte Carlo](https://mlsanity.com/paper/2309.12664/) (2023)
- [iMTSP: Solving Min-Max Multiple Traveling Salesman Problem with Imperative Learning](https://mlsanity.com/paper/2405.00285/) (2024)
- [Divide-and-Conquer Fusion](https://mlsanity.com/paper/2110.07265/) (2021)
- [Benchmarking Bayesian neural networks and evaluation metrics for regression tasks](https://mlsanity.com/paper/2206.06779/) (2022)
- [Centered plug-in estimation of Wasserstein distances](https://mlsanity.com/paper/2203.11627/) (2022)
- [Function Space Bayesian Pseudocoreset for Bayesian Neural Networks](https://mlsanity.com/paper/2310.17852/) (2023)
- [General bounds on the quality of Bayesian coresets](https://mlsanity.com/paper/2405.11780/) (2024)
- [Functional Stochastic Gradient MCMC for Bayesian Neural Networks](https://mlsanity.com/paper/2409.16632/) (2024)
- [Random Reshuffling for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2501.16055/) (2025)
- [Quantum Speedups for Sampling and Non-convex Optimization with Stochastic Oracles](https://mlsanity.com/paper/2504.03626/) (2025)
- [Improved Guarantees for Langevin Monte Carlo with Average Smoothness](https://mlsanity.com/paper/2605.31413/) (2026)
- [Quantifying the mini-batching error in Bayesian inference for Adaptive Langevin dynamics](https://mlsanity.com/paper/2105.10347/) (2021)
- [Scalable and Calibrated Sampling for Bayesian Generalized Linear Mixed Model via Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/2403.03007/) (2024)
- [Robust Stochastic Gradient Posterior Sampling with Lattice Based Discretisation](https://mlsanity.com/paper/2602.15925/) (2026)
- [Safe, Scalable, and Accurate Bayes Posterior Sampling for Large-Data Generalized Linear Mixed Models](https://mlsanity.com/paper/2604.26029/) (2026)
- [A New Framework for Variance-Reduced Hamiltonian Monte Carlo](https://mlsanity.com/paper/2102.04613/) (2021)
- [Preferential Subsampling for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2210.16189/) (2022)

## Abstract (arXiv)

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.