Solvable integration problems and optimal sample size selection

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 constantsthe 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.

Stability of doubly-intractable distributions

“…Furthermore, in Section 2.3 we only considered the expected difference of distributions. In the light of [6] and also [7,8] statements of the type "small error with high probability" are desirable. In particular, an investigation concerning the approximation of functions by Monte Carlo recovery algorithms seems to be a challenging and very interesting task.…”

Optimal confidence for Monte Carlo integration of smooth functions

“…Our lower bounds do hold for this type of algorithms, but the upper bounds we present are based on non-adaptive methods. For simplicity, in this paper we restrict to methods with fixed cardinality n. In general, the number of function values an algorithm collects might be random and even depend on the input, see for instance [8,12,16]. Let us mention here that our auxiliary lemmas on lower bounds, Lemma 2.1 and 2.2, would then still hold with slightly worse constants.…”

“…First, we consider Hölder classes C β ([0, 1] d ) with smoothness β ∈ (0, 1], see (16). Compare also [4] for the result in terms of the root mean squared error.…”

“…where · W is the (semi-)norm of W. A method with this property of guaranteeing a small (absolute) error ε > 0 with confidence 1 − δ ∈ (0, 1) (or uncertainty δ) for inputs from the unit ball B W is called (ε, δ)-approximating in W, see also [16]. We also consider the n-th minimal probabilistic Monte Carlo error at uncertainty δ, defined by e MC prob (n, δ, W) := inf {ε > 0 | ∃ (ε, δ)-approximating algorithm A n in W} .…”

“…For this reason the probabilistic criterion is frequently used in statistics, see for example [8,11,12,13]. Furthermore, there are numerical problems which can be solved with respect to the probabilistic (ε, δ)-criterion but the mean error is unbounded, see [16]. In other words, this criterion seems to be the right one for the concept of solvability.…”