In particular, in Appendix E.1 we show the Table 2 provides the notations used throughout the paper. The precise expressions are given throughout the Appendix in stated sections. In this section, we provide the proof of Theorem 1 with precise expressions. Combining (26) and (27) gives the self-normalized estimation error bound state in the theorem.D.2 Frobenius Norm Bound on Finite Sample Estimation Error of (10) Using this result, we have σ It represents the effect of noises in the system on the outputs. E.1 Persistence of Excitation in Warm-up Recall the state-space form of the system, x Using Weyl's inequality, during the warm-up period with probability 1 δ, we have σ Now consider when the underlying system is known.

We study the problem of adaptive control in partially observable linear dynamical systems. We propose a novel algorithm, adaptive control online learning algorithm (AdaptOn), which efficiently explores the environment, estimates the system dynamics episodically and exploits these estimates to design effective controllers to minimize the cumulative costs. Through interaction with the environment, AdaptOn deploys online convex optimization to optimize the controller while simultaneously learning the system dynamics to improve the accuracy of controller updates. We show that when the cost functions are strongly convex, after $T$ times step of agent-environment interaction, AdaptOn achieves regret upper bound of $\text{polylog}\left(T\right)$. To the best of our knowledge, AdaptOn is the first algorithm which achieves $\text{polylog}\left(T\right)$ regret in adaptive control of unknown partially observable linear dynamical systems which includes linear quadratic Gaussian (LQG) control.