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The problem of estimating the $\mathcal{H}_\infty$-norm of an LTI system fromnoisy input/output measurements has attracted recent attention as analternative to parameter identification for bounding unmodeled dynamics inrobust control. In this paper, we study lower bounds for$\mathcal{H}_\infty$-norm estimation under a query model where at eachiteration the algorithm chooses a bounded input signal and receives theresponse of the chosen signal corrupted by white noise. We prove that when theunderlying system is an FIR filter, $\mathcal{H}_\infty$-norm estimation is nomore efficient than model identification for passive sampling. For activesampling, we show that norm estimation is at most a factor of $\log{r}$ moresample efficient than model identification, where $r$ is the length of thefilter. We complement our theoretical results with experiments whichdemonstrate that a simple non-adaptive estimator of the norm is competitivewith state-of-the-art adaptive norm estimation algorithms.

Minimax Lower Bounds for $\mathcal{H}_\infty$-Norm Estimation