Minimax Number of Strata for Online Stratified Sampling given Noisy Samples

We consider the problem of online stratified sampling for Monte Carlo integration of a function given a finite budget of $n$ noisy evaluations to the function. More precisely we focus on the problem of choosing the number of strata $K$ as a function of the budget $n$. We provide asymptotic and finite-time results on how an oracle that has access to the function would choose the partition optimally. In addition we prove a \textit{lower bound} on the learning rate for the problem of stratified Monte-Carlo. As a result, we are able to state, by improving the bound on its performance, that algorithm MC-UCB, defined in \citep{MC-UCB}, is minimax optimal both in terms of the number of samples n and the number of strata K, up to a $\sqrt{\log(nK)}$. This enables to deduce a minimax optimal bound on the difference between the performance of the estimate outputted by MC-UCB, and the performance of the estimate outputted by the best oracle static strategy, on the class of H\"older continuous functions, and upt to a $\sqrt{\log(n)}$.