Towards a Dimension-Free Understanding of Adaptive Linear Control
This addresses a foundational problem in control theory for systems with high-dimensional states, offering a dimension-free approach that is novel for infinite-dimensional settings.
The paper tackles adaptive control of linear quadratic regulators in high or infinite dimensions, showing that sublinear regret is possible without bounding the state dimension, and provides the first regret bounds for infinite-dimensional systems with near-optimal scaling in finite cases.
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.