Instance-optimal PAC Algorithms for Contextual Bandits
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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 (ϵ,δ)-PAC setting: given a policy class Π the goal of the learner is to return a policy π∈Π whose expected reward is within ϵ of the optimal policy with probability greater than 1-δ. We characterize the first instance-dependent PAC sample complexity of contextual bandits through a quantity ρ_Π, and provide matching upper and lower bounds in terms of ρ_Π 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.
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