Breaking the Curse for Uniform Approximation in Hilbert Spaces via Monte Carlo Methods

TL;DR

This paper demonstrates that Monte Carlo methods can overcome the curse of dimensionality in uniform approximation problems within certain Hilbert spaces, achieving polynomial tractability where deterministic methods fail.

Contribution

It shows that for specific periodic tensor product spaces, Monte Carlo methods break the curse of dimensionality, providing a new approach for high-dimensional approximation problems.

Findings

Monte Carlo methods achieve polynomial tractability in certain Hilbert spaces.

Switching to randomized methods overcomes exponential growth in information needed.

Applicable to Korobov spaces with smoothness r > 1/2.

Abstract

We study the $L_{\infty}$-approximation of $d$-variate functions from Hilbert spaces via linear functionals as information. It is a common phenomenon in tractability studies that unweighted problems (with each dimension being equally important) suffer from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of information needed in order to solve a problem to within a given accuracy $\varepsilon > 0$ grows exponentially in $d$. We show that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness $r > 1/2$, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability, namely $n(\varepsilon,d) \preceq \varepsilon^{-2} \, d \, (1 + \log d)$. Similar benefits of Monte Carlo methods in terms of tractability have only been known for integration problems so far.