Black-Box Control for Linear Dynamical Systems

We consider the problem of black-box control: the task of controlling an unknown linear time-invariant dynamical system from a single trajectory without a stabilizing controller. Under the assumption that the system is controllable, we give the first {\it efficient} algorithm that is capable of attaining sublinear regret under the setting of online nonstochastic control. This resolves an open problem since the work of Abbasi-Yadkori and Szepesvari(2011) on the stochastic LQR problem, and in a more challenging setting that allows for adversarial perturbations and adversarially chosen changing convex loss functions. We give finite-time regret bounds for our algorithm on the order of $2^{poly(d)} + \tilde{O}(poly(d) T^{2/3})$ for general nonstochastic control, and $2^{poly(d)} + \tilde{O}(poly(d) \sqrt{T})$ for black-box LQR. To complete the picture, we investigate the complexity of the online black-box control problem and give a matching regret lower bound of $2^{\Omega(d)}$, showing that the exponential cost is inevitable. This lower bound holds even in the noiseless setting, and applies to any, randomized or deterministic, black-box control method.