Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms

TL;DR

This paper establishes fundamental lower error bounds for randomized algorithms tackling high-dimensional integration, highlighting the effectiveness of multilevel and changing dimension methods under different cost models.

Contribution

It introduces the first non-trivial lower error bounds for randomized algorithms in infinite-dimensional integration with weighted Hilbert spaces, and analyzes their optimality.

Findings

Multilevel algorithms perform well under the first cost model.

Changing dimension algorithms benefit from the second cost model.

Existing algorithms can achieve near-optimal convergence rates.

Abstract

We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on weighted Hilbert spaces with an underlying anchored decomposition and arbitrary weights. We focus on randomized algorithms and the randomized worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points. Multilevel algorithms behave very well with respect to the first cost model, while changing dimension algorithms and also dimension-wise quadrature methods, which are based on a similar idea, can take advantage of the more generous second cost model. We prove the first non-trivial lower error bounds for randomized algorithms in these cost models and demonstrate their quality in the case of product weights. In particular, we show that the randomized changing dimension algorithms provided in [L. Plaskota, G. W. Wasilkowski, J. Complexity 27 (2011), 505--518] achieve convergence rates arbitrarily close to the optimal convergence rate.