Minimax number of strata for online stratified sampling: The case of noisy samples

We consider online stratied sampling for Monte Carlo estimation of the integral of a function given a nite budget n of noisy evaluations to the function. In this paper we address the problem of choosing the best number K of strata as a function of n. A large K provides a high quality stratication where an accurate estimate of the integral of f could be computed by an optimal oracle allocation if the variances within each stratum were known. However the performance of an adaptive allocation (which does not know the variance within the strata) compared to the oracle one deteriorates with K. This denes a trade-o between the stratication quality and the pseudo-regret of an adaptive strategy.