Making Non-Stochastic Control (Almost) as Easy as Stochastic

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.