Any-Time Regret-Guaranteed Algorithm for Control of Linear Quadratic Systems
We propose a computationally efficient algorithm that achieves anytime regret of order $\mathcal{O}(\sqrt{t})$, with explicit dependence on the system dimensions and on the solution of the Discrete Algebraic Riccati Equation (DARE). Our approach builds on the SDP-based framework of \cite{cohen2019learning}, using an appropriately tuned regularization and a sufficiently accurate initial estimate to construct confidence ellipsoids for control design. A carefully designed input-perturbation mechanism is incorporated to ensure anytime performance. We develop two variants of the algorithm. The first enforces a notion of strong sequential stability, requiring each policy to be stabilizing and successive policies to remain close. However, enforcing this notion results in a suboptimal regret scaling. The second removes the sequential-stability requirement and instead requires only that each generated policy be stabilizing. Closed-loop stability is then preserved through a dwell-time-inspired policy-update rule, adapting ideas from switched-systems control to carefully balance exploration and exploitation. This class of algorithms also addresses key shortcomings of most existing approaches including certainty-equivalence-based methods which typically guarantee stability only in the Lyapunov sense and lack explicit uniform high-probability bounds on the state trajectory expressed in system-theoretic terms. Our analysis explicitly characterizes the trade-off between state amplification and regret, and shows that partially relaxing the sequential-stability requirement yields optimal regret. Finally, our method eliminates the need for any a priori bound on the norm of the DARE solution, an assumption required by all existing computationally efficient optimism in the face of uncertainty (OFU) based algorithms, and thereby removes the reliance of regret guarantees on such external inputs.