Logarithmic regret for episodic continuous-time linear-quadratic reinforcement learning over a finite-time horizon
This provides a theoretical guarantee for reinforcement learning in continuous-time control systems with unknown parameters, though it is incremental as it builds on existing LQ frameworks.
The paper tackles the problem of episodic continuous-time linear-quadratic reinforcement learning with unknown coefficients over a finite-time horizon, achieving a logarithmic regret bound of order O((ln M)(ln ln M)) for a least-squares algorithm based on continuous-time observations.
We study finite-time horizon continuous-time linear-quadratic reinforcement learning problems in an episodic setting, where both the state and control coefficients are unknown to the controller. We first propose a least-squares algorithm based on continuous-time observations and controls, and establish a logarithmic regret bound of order $O((\ln M)(\ln\ln M))$, with $M$ being the number of learning episodes. The analysis consists of two parts: perturbation analysis, which exploits the regularity and robustness of the associated Riccati differential equation; and parameter estimation error, which relies on sub-exponential properties of continuous-time least-squares estimators. We further propose a practically implementable least-squares algorithm based on discrete-time observations and piecewise constant controls, which achieves similar logarithmic regret with an additional term depending explicitly on the time stepsizes used in the algorithm.