Uniform approximation of vectors using adaptive randomized information
Robert J. Kunsch, Marcin Wnuk
TL;DR
The paper addresses the problem of uniformly approximating the embedding $\ell_p^m \to \ell_\infty^m$ for $1 \le p \le 2$ using randomized adaptive information consisting of arbitrary linear functionals. It develops a toolkit—hashing to create buckets, bucket selection to isolate promising buckets, and accelerated shrinking to spot heavy hitters—that enables adaptive identification of large coordinates and efficient approximation. The authors prove that the uniform error can be achieved with information cost $\mathrm{card}(A_p^{\varepsilon,\delta}) \preceq (\log\delta^{-1}+\log\varepsilon^{-1}+\log\log m)\varepsilon^{-p}$ and obtain a bound on the $n$-th minimal error $e^{ran}(n, \ell_p^m \hookrightarrow \ell_\infty^m) \preceq \min\{1,(\log n+\log\log m)/n)^{1/p}\}$, illustrating the advantages of adaptivity over non-adaptive Monte Carlo in this setting. A central contribution is showing that adaptivity can yield gaps up to order $n$ (up to logarithmic factors) between adaptive and non-adaptive schemes for linear problems, including explicit constructions and corollaries. Overall, the work advances information-based complexity for finite-dimensional sequence space embeddings and clarifies the practical impact of adaptivity in randomized linear approximation.
Abstract
We study approximation of the embedding $\ell_p^m \rightarrow \ell_{\infty}^m$, $1 \leq p \leq 2$, based on randomized adaptive algorithms that use arbitrary linear functionals as information on a problem instance. We show upper bounds for which the complexity $n$ exhibits only a $(\log\log m)$-dependence. Our results for $p=1$ lead to an example of a gap of order $n$ (up to logarithmic factors) for the error between best adaptive and non-adaptive Monte Carlo methods. This is the largest possible gap for linear problems.