Optimal Dynamic Regret in LQR Control
Abstract
We consider the problem of nonstochastic control with a sequence of quadratic losses, i.e., LQR control. We provide an efficient online algorithm that achieves an optimal dynamic (policy) regret of $\tilde{O}(n^{1/3} \mathcal{TV}(M_{1:n}^{2/3} \vee 1)$, where $\mathcal{TV}(M_{1:n})$ is the total variation of any oracle sequence of \emph{Disturbance Action} policies parameterized by $M_1,...,M_n$ --- chosen in hindsight to cater to unknown nonstationarity. The rate improves the best known rate of $\tilde{O}(\sqrt{n (\mathcal{TV}(M_{1:n})+1)} )$ for general convex losses and is information-theoretically optimal for LQR. Main technical components include the reduction of LQR to online linear regression with delayed feedback due to Foster & Simchowitz 2020, as well as a new \emph{proper} learning algorithm with an optimal $\tilde{O}(n^{1/3})$ dynamic regret on a family of "minibatched'' quadratic losses, which could be of independent interest.
BibTeX
@inproceedings{
baby2022optimal,
title={Optimal Dynamic Regret in {LQR} Control},
author={Dheeraj Baby and Yu-Xiang Wang},
booktitle={Advances in Neural Information Processing Systems},
editor={Alice H. Oh and Alekh Agarwal and Danielle Belgrave and Kyunghyun Cho},
year={2022},
url={https://openreview.net/forum?id=D0aqV81d0_k}
}