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Unimodal Bandits with Continuous Arms

We consider stochastic bandit problems with a continuous set of arms and where the expected reward is a continuous and unimodal function of the arm. For these problems, we propose the Stochastic Polychotomy (SP) algorithms, and derive finite-time upper bounds on their regret and optimization error. We show that, for a class of reward functions, the SP algorithm achieves a regret and an optimization error with optimal scalings, i.e., O( √ T ) and O(1/ √ T ) (up to a logarithmic factor), respectively.