Toward Optimal Stratification for Stratified Monte-Carlo Integration

TL;DR

This paper presents an adaptive stratified sampling algorithm for Monte Carlo integration that dynamically refines the domain partition and allocates samples optimally, balancing detail and efficiency.

Contribution

It introduces a novel algorithm that adaptively selects the best partition and sample allocation for noisy function integration within a fixed budget.

Findings

The algorithm effectively balances partition refinement and sample allocation.

It achieves near-optimal sample distribution across partitions.

The method improves integration accuracy over non-adaptive approaches.

Abstract

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.