  This paper establishes problem-specific sample complexity lower bounds for
linear system identification problems. The sample complexity is defined in the
PAC framework: it corresponds to the time it takes to identify the system
parameters with prescribed accuracy and confidence levels. By problem-specific,
we mean that the lower bound explicitly depends on the system to be identified
(which contrasts with minimax lower bounds), and hence really captures the
identification hardness specific to the system. We consider both uncontrolled
and controlled systems. For uncontrolled systems, the lower bounds are valid
for any linear system, stable or not, and only depend of the system finite-time
controllability gramian. A simplified lower bound depending on the spectrum of
the system only is also derived. In view of recent finitetime analysis of
classical estimation methods (e.g. ordinary least squares), our sample
complexity lower bounds are tight for many systems. For controlled systems, our
lower bounds are not as explicit as in the case of uncontrolled systems, but
could well provide interesting insights into the design of control policy with
minimal sample complexity.
