  In this paper, we investigate when system identification is statistically
easy or hard, in the finite sample regime. Statistically easy to learn linear
system classes have sample complexity that is polynomial with the system
dimension. Most prior research in the finite sample regime falls in this
category, focusing on systems that are directly excited by process noise.
Statistically hard to learn linear system classes have worst-case sample
complexity that is at least exponential with the system dimension, regardless
of the identification algorithm. Using tools from minimax theory, we show that
classes of linear systems can be hard to learn. Such classes include, for
example, under-actuated or under-excited systems with weak coupling among the
states. Having classified some systems as easy or hard to learn, a natural
question arises as to what system properties fundamentally affect the hardness
of system identifiability. Towards this direction, we characterize how the
controllability index of linear systems affects the sample complexity of
identification. More specifically, we show that the sample complexity of
robustly controllable linear systems is upper bounded by an exponential
function of the controllability index. This implies that identification is easy
for classes of linear systems with small controllability index and potentially
hard if the controllability index is large. Our analysis is based on recent
statistical tools for finite sample analysis of system identification as well
as a novel lower bound that relates controllability index with the least
singular value of the controllability Gramian.
