  Recent literature has made much progress in understanding \emph{online LQR}:
a modern learning-theoretic take on the classical control problem in which a
learner attempts to optimally control an unknown linear dynamical system with
fully observed state, perturbed by i.i.d. Gaussian noise. It is now understood
that the optimal regret on time horizon $T$ against the optimal control law
scales as $\widetilde{\Theta}(\sqrt{T})$. In this paper, we show that the same
regret rate (against a suitable benchmark) is attainable even in the
considerably more general non-stochastic control model, where the system is
driven by \emph{arbitrary adversarial} noise (Agarwal et al. 2019). In other
words, \emph{stochasticity confers little benefit in online LQR}.
  We attain the optimal $\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the
dynamics are unknown to the learner, and $\mathrm{poly}(\log T)$ regret when
known, provided that the cost functions are strongly convex (as in LQR). Our
algorithm is based on a novel variant of online Newton step (Hazan et al.
2007), which adapts to the geometry induced by possibly adversarial
disturbances, and our analysis hinges on generic "policy regret" bounds for
certain structured losses in the OCO-with-memory framework (Anava et al. 2015).
Moreover, our results accomodate the full generality of the non-stochastic
control setting: adversarially chosen (possibly non-quadratic) costs, partial
state observation, and fully adversarial process and observation noise.
