  We study the control of an \emph{unknown} linear dynamical system under
general convex costs. The objective is minimizing regret vs. the class of
disturbance-feedback-controllers, which encompasses all stabilizing
linear-dynamical-controllers. In this work, we first consider the case of known
cost functions, for which we design the first polynomial-time algorithm with
$n^3\sqrt{T}$-regret, where $n$ is the dimension of the state plus the
dimension of control input. The $\sqrt{T}$-horizon dependence is optimal, and
improves upon the previous best known bound of $T^{2/3}$. The main component of
our algorithm is a novel geometric exploration strategy: we adaptively
construct a sequence of barycentric spanners in the policy space. Second, we
consider the case of bandit feedback, for which we give the first
polynomial-time algorithm with $poly(n)\sqrt{T}$-regret, building on Stochastic
Bandit Convex Optimization.
