  We consider the online control problem with an unknown linear dynamical
system in the presence of adversarial perturbations and adversarial convex loss
functions. Although the problem is widely studied in model-based control, it
remains unclear whether data-driven approaches, which bypass the system
identification step, can solve the problem. In this work, we present a novel
data-driven online adaptive control algorithm to address this online control
problem. Our algorithm leverages the behavioral systems theory to learn a
non-parametric system representation and then adopts a perturbation-based
controller updated by online gradient descent. We prove that our algorithm
guarantees an $\tmO(T^{2/3})$ regret bound with high probability, which matches
the best-known regret bound for this problem. Furthermore, we extend our
algorithm and performance guarantee to the cases with output feedback.
