Solvable Integration Problems and Optimal Sample Size Selection

TL;DR

This paper introduces an optimal algorithm for computing integrals or expectations with minimal cost using i.i.d. samples, providing bounds and conditions for solvability based on variance or $p$-moment estimates.

Contribution

The paper presents a new adaptive sampling algorithm with proven optimality bounds for integral and expectation estimation based on variance or $p$-moment estimates.

Findings

The algorithm guarantees small absolute error with high probability.

Lower bounds demonstrate the optimality of the method in terms of accuracy and confidence.

Adaptive stopping rules ensure finite expected cost for each input.

Abstract

We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a sample size based on a variance estimation, or -- more generally -- based on an estimation of a (central absolute) $p$-moment. That way one can guarantee a small absolute error with high probability, the problem is thus called solvable. The expected cost of the method depends on the $p$-moment of the random variable, which can be arbitrarily large. In order to prove the optimality of our algorithm we also provide lower bounds. These bounds apply not only to methods based on i.i.d. samples but also to general randomized algorithms. They show that -- up to constants -- the cost of the algorithm is optimal in terms of accuracy, confidence level, and norm of the particular input random variable. Since the considered classes of random variables or integrands are very large, the worst case cost would be infinite. Nevertheless one can define adaptive stopping rules such that for each input the expected cost is finite. We contrast these positive results with examples of integration problems that are not solvable.