  We consider adaptive control of the Linear Quadratic Regulator (LQR), where
an unknown linear system is controlled subject to quadratic costs. Leveraging
recent developments in the estimation of linear systems and in robust
controller synthesis, we present the first provably polynomial time algorithm
that provides high probability guarantees of sub-linear regret on this problem.
We further study the interplay between regret minimization and parameter
estimation by proving a lower bound on the expected regret in terms of the
exploration schedule used by any algorithm. Finally, we conduct a numerical
study comparing our robust adaptive algorithm to other methods from the
adaptive LQR literature, and demonstrate the flexibility of our proposed method
by extending it to a demand forecasting problem subject to state constraints.
