On a Phase Transition of Regret in Linear Quadratic Control: The Memoryless Case

We consider an idealized version of adaptive control of a multiple input multiple output (MIMO) system without state. We demonstrate how rank deficient Fisher information in this simple memoryless problem leads to the impossibility of logarithmic rates of regret. Our analysis rests on a version of the Cramèr-Rao inequality that takes into account possible ill-conditioning of Fisher information and a pertubation result on the corresponding singular subspaces. This is used to define a sufficient condition, which we term uniformativeness, for regret to be at least order square root in the samples.