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.

翻译：我们研究了线性二次调节器在高度或甚至无限维度系统中的适应性控制问题。我们证明,尽管亚线性遗憾需要有限的次线性输入,但系统的环境状态层面不需要为进行在线控制而受约束。我们为持有无限维度系统的LQR提供了第一个遗憾界限,用更自然的复杂问题概念取代对环境层面的依赖。我们的保障产生于一种新颖的扰动,约束于确定性等同,在估计系统参数时,以预测误差为尺度,而不需要在操作员规范等更为严格的措施下进行一致的参数恢复。当我们专门使用有限维度设置时,我们的界限恢复到接近最佳维度和时间范围依赖。