  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.
