Foundations of Safe Online Reinforcement Learning in the Linear Quadratic Regulator: $\sqrt{T}$-Regret

Understanding how to efficiently learn while adhering to safety constraints is essential for using online reinforcement learning in practical applications. However, proving rigorous regret bounds for safety-constrained reinforcement learning is difficult due to the complex interaction between safety, exploration, and exploitation. In this work, we seek to establish foundations for safety-constrained reinforcement learning by studying the canonical problem of controlling a one-dimensional linear dynamical system with unknown dynamics. We study the safety-constrained version of this problem, where the state must with high probability stay within a safe region, and we provide the first safe algorithm that achieves regret of $\tilde{O}_T(\sqrt{T})$. Furthermore, the regret is with respect to the baseline of truncated linear controllers, a natural baseline of non-linear controllers that are well-suited for safety-constrained linear systems. In addition to introducing this new baseline, we also prove several desirable continuity properties of the optimal controller in this baseline. In showing our main result, we prove that whenever the constraints impact the optimal controller, the non-linearity of our controller class leads to a faster rate of learning than in the unconstrained setting.