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

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.