Context-lumpable stochastic bandits

We consider a contextual bandit problem with $S $ contexts and $K $ actions. In each round $t=1,2,\dots$ the learnerobserves 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 $r\le \min(S ,K)$ 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 $\epsilon$-optimal policy after using at most $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +K )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r ^3(S +K )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O{\sqrt{\text{poly}(r)(S+K)T}}$ 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.