A Note on Monte Carlo Integration in High Dimensions

TL;DR

This paper investigates the performance of Monte Carlo integration in high dimensions, showing that its efficiency can vary widely and providing bounds and confidence intervals that are valid regardless of sample size.

Contribution

It introduces non-asymptotic bounds and confidence intervals for Monte Carlo integration in high dimensions, revealing variable convergence rates.

Findings

Sample complexity can vary from polynomial to exponential in high dimensions.

Monte Carlo integration can achieve arbitrarily fast or slow convergence rates.

Non-asymptotic confidence intervals are valid regardless of sample size.

Abstract

Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive non-asymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate vary between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behaviour of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain non-asymptotic confidence intervals for Monte Carlo estimate which are valid regardless of the number of points sampled.