Thompson sampling for linear quadratic mean-field teams

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 $\widetilde {\mathcal{O}}\left({{{\left| M \right|}^{1.5}}\sqrt T }\right)$ irrespective of the total number of agents, where the $\widetilde {\mathcal{O}}$ notation hides logarithmic factors in T . We present detailed numerical experiments to illustrate the salient features of the proposed algorithm.