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We consider adaptive control of the Linear Quadratic Regulator (LQR), wherean unknown linear system is controlled subject to quadratic costs. Leveragingrecent developments in the estimation of linear systems and in robustcontroller synthesis, we present the first provably polynomial time algorithmthat provides high probability guarantees of sub-linear regret on this problem.We further study the interplay between regret minimization and parameterestimation by proving a lower bound on the expected regret in terms of theexploration schedule used by any algorithm. Finally, we conduct a numericalstudy comparing our robust adaptive algorithm to other methods from theadaptive LQR literature, and demonstrate the flexibility of our proposed methodby extending it to a demand forecasting problem subject to state constraints.

Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator