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

TL;DR

This work analyzes randomized approximation of the identity embedding from $\ell_1^m$ to $\ell_\infty^m$, contrasting non-adaptive and adaptive information models. It establishes a non-adaptive Monte Carlo lower bound that grows with the ambient dimension $m$, showing non-compact operators cannot be approximated non-adaptively; it also proves a universal non-compactness result using Jessen–Pettis type principles. In contrast, adaptive randomized methods achieve significantly better rates, with upper bounds of order $\sqrt{\log\log(m/n)/n}$ for large $m$, illustrating a substantial gap between adaptive and non-adaptive information. The results connect information-based complexity, Gaussian-mixture inputs, and sparse-recovery techniques to elucidate when adaptivity and nonlinearity yield meaningful advantages in randomized approximation of linear operators.

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 $n^{1/2} ( \log n)^{-1/2}$.

Theorems & Definitions (27)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof