On Uninformative Optimal Policies in Adaptive LQR with Unknown B-Matrix

Understand

This paper presents local asymptotic minimax regret lower bounds for adaptive Linear Quadratic Regulators (LQR).

• We consider affinely parametrized $B$-matrices and known $A$-matrices and aim to understand when logarithmic regret is impossible even in the presence of structural side information.
• After defining the intrinsic notion of an uninformative optimal policy in terms of a singularity condition for Fisher information we obtain local minimax regret lower bounds for such uninformative instances of LQR by appealing to van Trees' inequality (Bayesian Cram\'er-Rao) and a representation of regret in terms of a quadratic form (Bellman error).
• It is shown that if the parametrization induces an uninformative optimal policy, logarithmic regret is impossible and the rate is at least order square root in the time horizon.