  We study the problem of adaptive control of the linear quadratic regulator
for systems in very high, or even infinite dimension. We demonstrate that while
sublinear regret requires finite dimensional inputs, the ambient state
dimension of the system need not be bounded in order to perform online control.
We provide the first regret bounds for LQR which hold for infinite dimensional
systems, replacing dependence on ambient dimension with more natural notions of
problem complexity. Our guarantees arise from a novel perturbation bound for
certainty equivalence which scales with the prediction error in estimating the
system parameters, without requiring consistent parameter recovery in more
stringent measures like the operator norm. When specialized to finite
dimensional settings, our bounds recover near optimal dimension and time
horizon dependence.
