Neural Control Variates for Monte Carlo Variance Reduction

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.