Naive Exploration is Optimal for Online LQR

TL;DR

This paper proves that naive exploration with certainty equivalent control is optimal for online LQR, establishing tight bounds on regret that depend on system dimensions and ruling out polylogarithmic regret algorithms.

Contribution

It introduces new upper and lower bounds for online LQR regret, demonstrating the optimality of a simple exploration strategy and developing the self-bounding ODE method for Riccati equations.

Findings

Optimal regret scales as rac{rac{d_{u}^2 d_{x} T}

Lower bounds exclude polylogarithmic regret algorithms

Simple certainty equivalent control with exploration is optimal

Abstract

We consider the problem of online adaptive control of the linear quadratic regulator, where the true system parameters are unknown. We prove new upper and lower bounds demonstrating that the optimal regret scales as $\widetilde{\Theta}({\sqrt{d_{\mathbf{u}}^2 d_{\mathbf{x}} T}})$, where $T$ is the number of time steps, $d_{\mathbf{u}}$ is the dimension of the input space, and $d_{\mathbf{x}}$ is the dimension of the system state. Notably, our lower bounds rule out the possibility of a $\mathrm{poly}(\log{}T)$-regret algorithm, which had been conjectured due to the apparent strong convexity of the problem. Our upper bound is attained by a simple variant of $\textit{{certainty equivalent control}}$, where the learner selects control inputs according to the optimal controller for their estimate of the system while injecting exploratory random noise. While this approach was shown to achieve $\sqrt{T}$-regret by (Mania et al. 2019), we show that if the learner continually refines their estimates of the system matrices, the method attains optimal dimension dependence as well. Central to our upper and lower bounds is a new approach for controlling perturbations of Riccati equations called the $\textit{self-bounding ODE method}$, which we use to derive suboptimality bounds for the certainty equivalent controller synthesized from estimated system dynamics. This in turn enables regret upper bounds which hold for $\textit{any stabilizable instance}$ and scale with natural control-theoretic quantities.