  We study the performance of the certainty equivalent controller on Linear
Quadratic (LQ) control problems with unknown transition dynamics. We show that
for both the fully and partially observed settings, the sub-optimality gap
between the cost incurred by playing the certainty equivalent controller on the
true system and the cost incurred by using the optimal LQ controller enjoys a
fast statistical rate, scaling as the square of the parameter error. To the
best of our knowledge, our result is the first sub-optimality guarantee in the
partially observed Linear Quadratic Gaussian (LQG) setting. Furthermore, in the
fully observed Linear Quadratic Regulator (LQR), our result improves upon
recent work by Dean et al. (2017), who present an algorithm achieving a
sub-optimality gap linear in the parameter error. A key part of our analysis
relies on perturbation bounds for discrete Riccati equations. We provide two
new perturbation bounds, one that expands on an existing result from
Konstantinov et al. (1993), and another based on a new elementary proof
strategy.
