Optimistic Online LQR via Intrinsic Rewards

Optimism in the face of uncertainty is a popular approach to balance exploration and exploitation in reinforcement learning. Here, we consider the online linear quadratic regulator (LQR) problem, i.e., to learn the LQR corresponding to an unknown linear dynamical system by adapting the control policy online based on closed-loop data collected during operation. In this work, we propose Intrinsic Rewards LQR (IR-LQR), an optimistic online LQR algorithm that applies the idea of intrinsic rewards originating from reinforcement learning and the concept of variance regularization to promote uncertainty-driven exploration. IR-LQR retains the structure of a standard LQR synthesis problem by only modifying the cost function, resulting in an intuitively pleasing, simple, computationally cheap, and efficient algorithm. This is in contrast to existing optimistic online LQR formulations that rely on more complicated iterative search algorithms or solve computationally demanding optimization problems. We show that IR-LQR achieves the optimal worst-case regret rate of $\sqrt{T}$, and compare it to various state-of-the-art online LQR algorithms via numerical experiments carried out on an aircraft pitch angle control and an unmanned aerial vehicle example.