  This paper studies a class of partially observed Linear Quadratic Gaussian
(LQG) problems with unknown dynamics. We establish an end-to-end sample
complexity bound on learning a robust LQG controller for open-loop stable
plants. This is achieved using a robust synthesis procedure, where we first
estimate a model from a single input-output trajectory of finite length,
identify an H-infinity bound on the estimation error, and then design a robust
controller using the estimated model and its quantified uncertainty. Our
synthesis procedure leverages a recent control tool called Input-Output
Parameterization (IOP) that enables robust controller design using convex
optimization. For open-loop stable systems, we prove that the LQG performance
degrades linearly with respect to the model estimation error using the proposed
synthesis procedure. Despite the hidden states in the LQG problem, the achieved
scaling matches previous results on learning Linear Quadratic Regulator (LQR)
controllers with full state observations.
