  The problem of estimating the $\mathcal{H}_\infty$-norm of an LTI system from
noisy input/output measurements has attracted recent attention as an
alternative to parameter identification for bounding unmodeled dynamics in
robust control. In this paper, we study lower bounds for
$\mathcal{H}_\infty$-norm estimation under a query model where at each
iteration the algorithm chooses a bounded input signal and receives the
response of the chosen signal corrupted by white noise. We prove that when the
underlying system is an FIR filter, $\mathcal{H}_\infty$-norm estimation is no
more efficient than model identification for passive sampling. For active
sampling, we show that norm estimation is at most a factor of $\log{r}$ more
sample efficient than model identification, where $r$ is the length of the
filter. We complement our theoretical results with experiments which
demonstrate that a simple non-adaptive estimator of the norm is competitive
with state-of-the-art adaptive norm estimation algorithms.
