Randomized approximation of summable sequences — adaptive and non-adaptive

We prove lower bounds for the randomized approximation of the embedding ${\ell }_1^m\hookrightarrow {\ell }_{\infty }^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N\in R^{n\times m}$. These lower bounds reflect the increasing difficulty of the problem for $m\to \infty$, namely, a term $\sqrt{logm}$ 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 $(loglogm)$-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 $n^{1/2}{(logn)}^{-1/2}$.