# Minimax Number of Strata for Online Stratified Sampling given Noisy Samples

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1 SEQUEL - Sequential Learning LIFL - Laboratoire d-Informatique Fondamentale de Lille, Inria Lille - Nord Europe, LAGIS - Laboratoire d-Automatique, Génie Informatique et Signal

Abstract : 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 $\sqrt{\lognK}$. 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 Hölder continuous functions, and upt to a $\sqrt{\logn}$.

Keywords : Online learning stratified sampling Monte Carlo integration regret bounds

Autor: Alexandra Carpentier - Rémi Munos -

Fuente: https://hal.archives-ouvertes.fr/

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