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TWe establish regret lower bounds for adaptively controlling an unknownlinear Gaussian system with quadratic costs. We combine ideas from experimentdesign, estimation theory and a perturbation bound of certain informationmatrices to derive regret lower bounds exhibiting scaling on the order ofmagnitude $\sqrt{T}$ in the time horizon $T$. Our bounds accurately capture therole of control-theoretic parameters and we are able to show that systems thatare hard to control are also hard to learn to control; when instantiated tostate feedback systems we recover the dimensional dependency of earlier workbut with improved scaling with system-theoretic constants such as system costsand Gramians. Furthermore, we extend our results to a class of partiallyobserved systems and demonstrate that systems with poor observability structurealso are hard to learn to control.

Regret Lower Bounds for Learning Linear Quadratic Gaussian Systems