Learning Decentralized Linear Quadratic Regulators with $\sqrt{T}$ Regret

We propose an online learning algorithm that adaptively designs a decentralized linear quadratic regulator when the system model is unknown a priori and new data samples from a single system trajectory become progressively available. The algorithm uses a disturbance-feedback representation of state-feedback controllers coupled with online convex optimization with memory and delayed feedback. Under the assumption that the system is stable or given a known stabilizing controller, we show that our controller enjoys an expected regret that scales as $\sqrt{T}$ with the time horizon $T$ for the case of partially nested information pattern. For more general information patterns, the optimal controller is unknown even if the system model is known. In this case, the regret of our controller is shown with respect to a linear sub-optimal controller. We validate our theoretical findings using numerical experiments.