  We introduce a new algorithm for online linear-quadratic control in a known
system subject to adversarial disturbances. Existing regret bounds for this
setting scale as $\sqrt{T}$ unless strong stochastic assumptions are imposed on
the disturbance process. We give the first algorithm with logarithmic regret
for arbitrary adversarial disturbance sequences, provided the state and control
costs are given by known quadratic functions. Our algorithm and analysis use a
characterization for the optimal offline control law to reduce the online
control problem to (delayed) online learning with approximate advantage
functions. Compared to previous techniques, our approach does not need to
control movement costs for the iterates, leading to logarithmic regret.
