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Monte Carlo (MC) sampling algorithms are an extremely widely-used techniqueto estimate expectations of functions f(x), especially in high dimensions.Control variates are a very powerful technique to reduce the error of suchestimates, but in their conventional form rely on having an accurateapproximation of f, a priori. Stacked Monte Carlo (StackMC) is a recentlyintroduced technique designed to overcome this limitation by fitting a controlvariate to the data samples themselves. Done naively, forming a control variateto the data would result in overfitting, typically worsening the MC algorithm'sperformance. StackMC uses in-sample / out-sample techniques to remove thisoverfitting. Crucially, it is a post-processing technique, requiring noadditional samples, and can be applied to data generated by any MC estimator.Our preliminary experiments demonstrated that StackMC improved the estimates ofexpectations when it was used to post-process samples produces by a "simplesampling" MC estimator. Here we substantially extend this earlier work. Weprovide an in-depth analysis of the StackMC algorithm, which we use toconstruct an improved version of the original algorithm, with lower estimationerror. We then perform experiments of StackMC on several additional kinds of MCestimators, demonstrating improved performance when the samples are generatedvia importance sampling, Latin-hypercube sampling and quasi-Monte Carlosampling. We also show how to extend StackMC to combine multiple fittingfunctions, and how to apply it to discrete input spaces x.

Reducing the error of Monte Carlo Algorithms by Learning Control Variates