Context-lumpable stochastic bandits

We consider a contextual bandit problem with S contexts and K actions. In each round t = 1, 2, ... the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into r ≤ min{S, K} groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an ε-optimal policy after using at most O^e(r(S + K)/ε²) samples with high probability and provide a matching Ω(r(S + K)/ε²) lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time T is bounded by (Equation presented). To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and O^e(ppoly(r)(S + K)T) minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.