# Logarithmic Regret for Episodic Continuous-Time Linear-Quadratic Reinforcement Learning over a Finite-Time Horizon

Matteo Basei, Xin Guo, Anran Hu, Yufei Zhang.

Year: 2022, Volume: 23, Issue: 178, Pages: 1−34

#### Abstract

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 magnitude $\mathcal{O}((\ln M)(\ln\ln M) )$, with $M$ being the number of learning episodes. The analysis consists of two components: 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.