Toward Optimal Stratification for Stratified Monte-Carlo Integration
This work addresses a specific challenge in Monte-Carlo integration for researchers in computational statistics or numerical analysis, presenting an incremental improvement by refining existing stratification methods.
The paper tackles the problem of adaptive stratified sampling for Monte-Carlo integration with a finite budget of noisy evaluations, developing an algorithm that simultaneously optimizes the partition of the domain and the allocation of samples to achieve an optimal trade-off.
We consider the problem of adaptive stratified sampling for Monte Carlo integration of a noisy function, given a finite budget n of noisy evaluations to the function. We tackle in this paper the problem of adapting to the function at the same time the number of samples into each stratum and the partition itself. More precisely, it is interesting to refine the partition of the domain in area where the noise to the function, or where the variations of the function, are very heterogeneous. On the other hand, having a (too) refined stratification is not optimal. Indeed, the more refined the stratification, the more difficult it is to adjust the allocation of the samples to the stratification, i.e. sample more points where the noise or variations of the function are larger. We provide in this paper an algorithm that selects online, among a large class of partitions, the partition that provides the optimal trade-off, and allocates the samples almost optimally on this partition.