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The theory of reinforcement learning currently suffers from a mismatchbetween its empirical performance and the theoretical characterization of itsperformance, with consequences for, e.g., the understanding of sampleefficiency, safety, and robustness. The linear quadratic regulator with unknowndynamics is a fundamental reinforcement learning setting with significantstructure in its dynamics and cost function, yet even in this setting there isa gap between the best known regret lower-bound of $\Omega_p(\sqrt{T})$ and thebest known upper-bound of $O_p(\sqrt{T}\,\text{polylog}(T))$. The contributionof this paper is to close that gap by establishing a novel regret upper-boundof $O_p(\sqrt{T})$. Our proof is constructive in that it analyzes the regret ofa concrete algorithm, and simultaneously establishes an estimation error boundon the dynamics of $O_p(T^{-1/4})$ which is also the first to match the rate ofa known lower-bound. The two keys to our improved proof technique are (1) amore precise upper- and lower-bound on the system Gram matrix and (2) aself-bounding argument for the expected estimation error of the optimalcontroller.

Rate-matching the regret lower-bound in the linear quadratic regulator with unknown dynamics