Neural Control Variates for Monte Carlo Variance Reduction

Ruosi Wan, Mingjun Zhong, Haoyi Xiong, Zhanxing Zhu*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingPublished conference contribution

2 Citations (Scopus)


In statistics and machine learning, approximation of an intractable integration is often achieved by using the unbiased Monte Carlo estimator, but the variances of the estimation are generally high in many applications. Control variates approaches are well-known to reduce the variance of the estimation. These control variates are typically constructed by employing predefined parametric functions or polynomials, determined by using those samples drawn from the relevant distributions. Instead, we propose to construct those control variates by learning neural networks to handle the cases when test functions are complex. In many applications, obtaining a large number of samples for Monte Carlo estimation is expensive, the adoption of the original loss function may result in severe overfitting when training a neural network. This issue was not reported in those literature on control variates with neural networks. We thus further introduce a constrained control variates with neural networks to alleviate the overfitting issue. We apply the proposed control variates to both toy and real data problems, including a synthetic data problem, Bayesian model evidence evaluation and Bayesian neural networks. Experimental results demonstrate that our method can achieve significant variance reduction compared to other methods.

Original languageEnglish
Title of host publicationMachine Learning and Knowledge Discovery in Databases
Subtitle of host publicationEuropean Conference, ECML PKDD 2019, Proceedings
EditorsUlf Brefeld, Elisa Fromont, Andreas Hotho, Arno Knobbe, Marloes Maathuis, Céline Robardet
Number of pages15
ISBN (Print)9783030461461
Publication statusPublished - 1 Jan 2020
EventEuropean Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, ECML PKDD 2019 - Wurzburg, Germany
Duration: 16 Sep 201920 Sep 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11907 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


ConferenceEuropean Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, ECML PKDD 2019


  • Control variates
  • Monte Carlo method
  • Neural networks
  • Variance reduction


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