Context-lumpable stochastic bandits
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We consider a contextual bandit problem with S contexts and A 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 ,A } 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(r (S +A )/ϵ^2) samples with high probability and provide a matching Ω(r (S +A )/ϵ^2) lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time T is bounded by O(√(r^3(S +A )T)). To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and O(√(poly(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.
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