Towards a Dimension-Free Understanding of Adaptive Linear Control

TL;DR

This paper establishes a dimension-free framework for adaptive linear control in high or infinite dimensions, providing regret bounds that depend on problem complexity rather than ambient dimension.

Contribution

It introduces the first regret bounds for infinite-dimensional LQR that replace ambient dimension dependence with natural complexity measures.

Findings

Regret bounds applicable to infinite-dimensional systems.

Dependence on problem complexity instead of ambient dimension.

Bounds recover near optimal dependence in finite-dimensional cases.

Abstract

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.