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

We consider online stratified sampling for Monte Carlo estimation of the integral of a function given a finite 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 stratification 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 defines a trade-off between the stratification quality and the pseudo-regret of an adaptive strategy. First we provide an improved pseudo-regret upper-bound of order Õ(K1/3n−4/3) for the adaptive allocation MC-UCB introduced in [1]. Then we prove a lower-bound on the pseudo-regret of same order, both in terms of K and n, up to a logarithmic factor. Finally we explain how to choose the best value of K given the budget n and deduce a tight minimax (on the class of Hölder continuous functions) optimal bound on the difference between the performance of the adaptive allocation MC-UCB, and the performance of the estimate returned by the optimal oracle strategy.