  We consider the problem of controlling an unknown linear quadratic Gaussian
(LQG) system consisting of multiple subsystems connected over a network. Our
goal is to minimize and quantify the regret (i.e. loss in performance) of our
strategy with respect to an oracle who knows the system model. Viewing the
interconnected subsystems globally and directly using existing LQG learning
algorithms for the global system results in a regret that increases
super-linearly with the number of subsystems. Instead, we propose a new
Thompson sampling based learning algorithm which exploits the structure of the
underlying network. We show that the expected regret of the proposed algorithm
is bounded by $\tilde{\mathcal{O}} \big( n \sqrt{T} \big)$ where $n$ is the
number of subsystems, $T$ is the time horizon and the
$\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms in $n$ and $T$.
Thus, the regret scales linearly with the number of subsystems. We present
numerical experiments to illustrate the salient features of the proposed
algorithm.
