Instance-optimal PAC Algorithms for Contextual Bandits

In the stochastic contextual bandit setting, regret-minimizing algorithms have been extensively researched, but their instance-minimizing best-arm identification counterparts remain seldom studied. In this work, we focus on the stochastic bandit problem in the $(\epsilon,\delta)$-$\textit{PAC}$ setting: given a policy class $\Pi$ the goal of the learner is to return a policy $\pi\in \Pi$ whose expected reward is within $\epsilon$ of the optimal policy with probability greater than $1-\delta$. We characterize the first $\textit{instance-dependent}$ PAC sample complexity of contextual bandits through a quantity $\rho{\Pi}$, and provide matching upper and lower bounds in terms of $\rho{\Pi}$ for the agnostic and linear contextual best-arm identification settings. We show that no algorithm can be simultaneously minimax-optimal for regret minimization and instance-dependent PAC for best-arm identification. Our main result is a new instance-optimal and computationally efficient algorithm that relies on a polynomial number of calls to an argmax oracle.