  We consider optimal control of an unknown multi-agent linear quadratic (LQ)
system where the dynamics and the cost are coupled across the agents through
the mean-field (i.e., empirical mean) of the states and controls. Directly
using single-agent LQ learning algorithms in such models results in regret
which increases polynomially with the number of agents. We propose a new
Thompson sampling based learning algorithm which exploits the structure of the
system model and show that the expected Bayesian regret of our proposed
algorithm for a system with agents of $|M|$ different types at time horizon $T$
is $\tilde{\mathcal{O}} \big( |M|^{1.5} \sqrt{T} \big)$ irrespective of the
total number of agents, where the $\tilde{\mathcal{O}}$ notation hides
logarithmic factors in $T$. We present detailed numerical experiments to
illustrate the salient features of the proposed algorithm.
