Instance-optimal PAC Algorithms for Contextual Bandits

Abstract

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

Tasks

Reproductions