Quantized Online LQR

We study online linear-quadratic regulation (LQR) with unknown dynamics under communication rate constraints. Classical networked control quantizes the plant state at every time step, requiring $O(T)$ total bits while injecting persistent quantization noise that limits control performance. We consider a setting where the plant observes its state locally and can estimate system dynamics via ordinary least squares, while a remote controller possesses knowledge of the control cost. Rather than quantizing the raw state, the plant transmits learned dynamics estimates over a rate-limited uplink, and the controller returns the optimal control policy so that the plant can compute actions locally using its superior state knowledge. We first prove a fundamental information-theoretic lower bound: any scheme achieving $O(T^\alpha)$ regret for $\alpha \in [1/2,1)$ compared to the optimal infinite horizon LQR controller that knows the true system dynamics must transmit at least $\Omega(\log T)$ bits. We then design the \textbf{Quantized Certainty Equivalent (QCE-LQR)} algorithm, which matches this bound. The resulting regret bound contains inflation factors $Q_{\mathrm{slow}}(\varrho)$ and $Q_{\mathrm{fast}}(\varrho)$ that vanish as the codebook resolution increases, smoothly recovering the unquantized baseline regret. Numerical experiments on four benchmark systems -- from a scalar unstable plant to a 24-parameter Boeing 747 lateral model -- confirm that a variant of QCE-LQR achieves regret comparable to an unquantized certainty equivalent controller over a horizon of $T=10{,}000$ steps.