Lower bounds for the number of random bits in Monte Carlo algorithms

TL;DR

This paper establishes lower bounds on the minimal errors and the number of random bits required for restricted Monte Carlo algorithms, extending previous results to adaptive settings and specific function spaces.

Contribution

It generalizes existing lower bounds to adaptive Monte Carlo algorithms and derives bounds for integration over Wiener space.

Findings

Lower bounds on minimal errors in restricted Monte Carlo algorithms.

Lower bounds on the number of random bits needed for integration of Lipschitz functions.

Extension of previous bounds to adaptive algorithm settings.

Abstract

We continue the study of restricted Monte Carlo algorithms in a general setting. Here we show a lower bound for minimal errors in the setting with finite restriction in terms of deterministic minimal errors. This generalizes a result of Heinrich, Novak, and Pfeiffer, 2004 to the adaptive setting. As a consequence, the lower bounds on the number of random bits from that paper also hold in this setting. We also derive a lower bound on the number of needed bits for integration of Lipschitz functions over the Wiener space, complementing a result of Giles, Hefter, Mayer, and Ritter, arXiv:1808.10623.