Simulation and the Monte Carlo Method (Wiley Series in Probability and Statistics)
Simulation and the Monte Carlo Method (Wiley Series in Probability and Statistics)
Asymptotically optimal allocation of stratified sampling with adaptive variance reduction by strata
ACM Transactions on Modeling and Computer Simulation (TOMACS)
| Hi-index | 0.00 |
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 numerical budget n. We provide asymptotic and finite-time results on how an oracle that knows the smoothness of the function would choose the number of strata optimally. In addition we prove a 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 [1, is minimax optimal both in terms of the number of samples n and the number of strata K, up to a log factor. This enables to deduce a minimax optimal bound on the difference between the performance of the estimate output by MC-UCB, and the performance of the estimate output by the best oracle static strategy, on the class of Hölder continuous functions, and up to a log factor.