  Inspired by the work of Tsiamis et al. \cite{tsiamis2022learning}, in this
paper we study the statistical hardness of learning to stabilize linear
time-invariant systems. Hardness is measured by the number of samples required
to achieve a learning task with a given probability. The work in
\cite{tsiamis2022learning} shows that there exist system classes that are hard
to learn to stabilize with the core reason being the hardness of
identification. Here we present a class of systems that can be easy to
identify, thanks to a non-degenerate noise process that excites all modes, but
the sample complexity of stabilization still increases exponentially with the
system dimension. We tie this result to the hardness of co-stabilizability for
this class of systems using ideas from robust control.
