Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

TL;DR

This paper investigates the efficiency of restricted Monte Carlo algorithms using random bits for approximating expectations of Gaussian measures in Hilbert spaces, revealing their competitiveness and optimality in certain settings.

Contribution

It provides asymptotic error rates for random bit Monte Carlo algorithms and links the problem to a novel quantization variant for probability measures.

Findings

Random bit algorithms are nearly as effective as full randomness methods.

Optimal multilevel random bit algorithms are identified as asymptotically optimal.

The results include error asymptotics for Gaussian measures and scalar SDEs.

Abstract

We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs.