# Theoretical guarantees for neural control variates in MCMC

arXiv: https://arxiv.org/abs/2304.01111 · HTML: https://mlsanity.com/paper/2304.01111/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 3 Apr 2023.

## Builds on (most cited first, 13 of 13)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions](https://mlsanity.com/paper/2007.05384/) (2020)
- [Approximating smooth functions by deep neural networks with sigmoid activation function](https://mlsanity.com/paper/2010.04596/) (2020)
- [PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units](https://mlsanity.com/paper/1909.05136/) (2019)
- [Regularized Zero-Variance Control Variates](https://mlsanity.com/paper/1811.05073/) (2018)
- [Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization](https://mlsanity.com/paper/2006.07487/) (2020)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [Empirical Variance Minimization with Applications in Variance Reduction and Optimal Control](https://mlsanity.com/paper/1712.04667/) (2017)
- [Variance reduction for Markov chains with application to MCMC](https://mlsanity.com/paper/1910.03643/) (2019)

## Built on by (most cited first, 4 of 4)

- [When can Regression-Adjusted Control Variates Help? Rare Events, Sobolev Embedding and Minimax Optimality](https://mlsanity.com/paper/2305.16527/) (2023)
- [Bayesian Numerical Integration with Neural Networks](https://mlsanity.com/paper/2305.13248/) (2023)
- [Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate](https://mlsanity.com/paper/2412.11257/) (2024)
- [Training neural control variates using correlated configurations](https://mlsanity.com/paper/2505.07719/) (2025)

## Abstract (arXiv)

In this paper, we propose a variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance. We focus on the particular case when control variates are represented as deep neural networks. We derive the optimal convergence rate of the asymptotic variance under various ergodicity assumptions on the underlying Markov chain. The proposed approach relies upon recent results on the stochastic errors of variance reduction algorithms and function approximation theory.