Regret Lower Bounds for Unbiased Adaptive Control of Linear Quadratic Regulators

We present lower bounds for the regret of adaptive control of the linear quadratic regulator. These are given in terms of problem specific expected regret lower bounds valid for unbiased policies linear in the state. Our approach is based on the insight that the adaptive control problem can, given our assumptions, be reduced to a sequential estimation problem. This enables the use of the Cramér-Rao information inequality which yields a scaling limit lower bound of logarithmic order. The bound features both information-theoretic and control-theoretic quantities. By leveraging existing results, we are able to show that the bound is tight in a special case.