Regret Analysis for Adaptive Linear-Quadratic Policies

In the classical problem of Linear-Quadratic (LQ) control, when the parameters of the system's dynamics are unknown, an adaptive policy is needed to learn those parameters and also plan a control action. The resulting trade-off between accurate parameter estimation (exploration) and effective control (exploitation) represents the main challenge in the area of adaptive control. Asymptotic approaches have been extensively studied in the literature, but there is a dearth of non-asymptotic results that in addition are rather incomplete. This study establishes high probability regret bounds for the aforementioned problem that are optimal up to logarithmic factors. The results on finite time analysis of the regret are obtained under very mild assumptions, requiring: (i) stabilizability of the system's dynamics, and (ii) limiting the degree of heaviness of the noise distribution. To establish such bounds, certain novel techniques are introduced to comprehensively address the probabilistic behavior of dependent random matrices with heavy-tailed distributions.