Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) control
are foundational and extensively researched problems in optimal control. We
investigate LQR and LQG problems with semi-adversarial perturbations and
time-varying adversarial bandit loss functions. The best-known sublinear regret
algorithm of~\cite{gradu2020non} has a T43β time horizon
dependence, and its authors posed an open question about whether a tight rate
of Tβ could be achieved. We answer in the affirmative, giving an
algorithm for bandit LQR and LQG which attains optimal regret (up to
logarithmic factors) for both known and unknown systems. A central component of
our method is a new scheme for bandit convex optimization with memory, which is
of independent interest