Adaptive Stratified Sampling for Monte-Carlo integration of Differentiable functions

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a differentiable function given a finite number of evaluations to the function. We construct a sampling scheme that samples more often in regions where the function oscillates more, while allocating the samples such that they are well spread on the domain (this notion shares similitude with low discrepancy). We prove that the estimate returned by the algorithm is almost as accurate as the estimate that an optimal oracle strategy (that would know the variations of the function everywhere) would return, and we provide a finite-sample analysis.