Randomized approximation of summable sequences -- adaptive and non-adaptive

TL;DR

This paper establishes lower bounds for randomized non-adaptive approximation of certain embeddings, showing adaptive methods can significantly outperform non-adaptive ones in terms of complexity and error.

Contribution

It provides the first lower bounds for non-adaptive randomized approximation of $oldsymbol{ ext{ell}_1^m ightarrow ext{ell}_ extinfty^m}$ embeddings, highlighting the advantage of adaptive methods.

Findings

Lower bounds grow with $oldsymbol{ ext{sqrt} ext{log} m}$ for non-adaptive methods.

Adaptive randomized methods have complexity depending on $oldsymbol{ ext{log} ext{log} m}$.

There exists a linear problem where adaptive methods outperform non-adaptive ones by a factor of $oldsymbol{n^{1/2} ( ext{log} n)^{-1/2}}$.

Abstract

We prove lower bounds for the randomized approximation of the embedding $\ell_1^m \rightarrow \ell_\infty^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N \in \mathbb{R}^{n \times m}$. These lower bounds reflect the increasing difficulty of the problem for $m \to \infty$, namely, a term $\sqrt{\log m}$ in the complexity $n$. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(\log\log m)$-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1/2} ( \log n)^{-1/2}$.