Uniform approximation of vectors using adaptive randomized information

We study approximation of the embedding ${\mathrm{\ell }}_p^m\to {\mathrm{\ell }}_{\mathrm{\infty }}^m$, $1\le p\le 2$, 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.