Motivated by a natural problem in online model selection with bandit information, we introduce and analyze a best arm identification problem in the rested bandit setting, wherein arm expected losses decrease with the number of times the arm has been played. The shape of the expected loss functions is similar across arms, and is assumed to be available up to unknown parameters that have to be learned on the fly. We define a novel notion of regret for this problem, where we compare to the policy that always plays the arm having the smallest expected loss at the end of the game. We analyze an arm elimination algorithm whose regret vanishes as the time horizon increases. The actual rate of convergence depends in a detailed way on the postulated functional form of the expected losses. Unlike known model selection efforts in the recent bandit literature, our algorithm exploits the specific structure of the problem to learn the unknown parameters of the expected loss function so as to identify the best arm as quickly as possible. We complement our analysis with a lower bound, indicating strengths and limitations of the proposed solution.
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