Almost Optimal Agnostic Control of Unknown Linear Dynamics

We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control problem: Bayesian control, in which we have a prior belief about $a$; bounded agnostic control, in which we have no prior belief about $a$ but we assume that $a$ belongs to a bounded set; and fully agnostic control, in which $a$ is allowed to be an arbitrary real number about which we have no prior belief. In the Bayesian variant, a control strategy is optimal if it minimizes a certain expected cost. In the agnostic variants, a control strategy is optimal if it minimizes a quantity called the worst-case regret. For the Bayesian and bounded agnostic variants above, we produce optimal control strategies. For the fully agnostic variant, we produce almost optimal control strategies, i.e., for any $\varepsilon>0$ we produce a strategy that minimizes the worst-case regret to within a multiplicative factor of $(1+\varepsilon)$.