Approximate Thompson Sampling for Learning Linear Quadratic Regulators with $O(\sqrt{T})$ Regret

We propose a novel Thompson sampling algorithm that learns linear quadratic regulators (LQR) with a Bayesian regret bound of $O(\sqrt{T})$. Our method leverages Langevin dynamics with a carefully designed preconditioner and incorporates a simple excitation mechanism. We show that the excitation signal drives the minimum eigenvalue of the preconditioner to grow over time, thereby accelerating the approximate posterior sampling process. Furthermore, we establish nontrivial concentration properties of the approximate posteriors generated by our algorithm. These properties enable us to bound the moments of the system state and attain an $O(\sqrt{T})$ regret bound without relying on the restrictive assumptions that are often used in the literature.