Finite Time Analysis of Stratified Sampling for Monte Carlo

We consider the problem of stratified sampling for Monte-Carlo integration. We model this problem in a multi-armed bandit setting, where the arms represent the strata, and the goal is to estimate a weighted average of the mean values of the arms. We propose a strategy that samples the arms according to an upper bound on their standard deviations and compare its estimation quality to an ideal allocation that would know the standard deviations of the arms. We provide two regret analyses: a distribution-dependent bound O(n^{-3/2}) that depends on a measure of the disparity of the arms, and a distribution-free bound O(n^{-4/3}) that does not. To the best of our knowledge, such a finite-time analysis is new for this problem.