On the complexity of All ε-Best Arms Identification
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We consider the problem introduced by <cit.> of identifying all the ε-optimal arms in a finite stochastic multi-armed bandit with Gaussian rewards. In the fixed confidence setting, we give a lower bound on the number of samples required by any algorithm that returns the set of ε-good arms with a failure probability less than some risk level δ. This bound writes as T_ε^*(μ)log(1/δ), where T_ε^*(μ) is a characteristic time that depends on the vector of mean rewards μ and the accuracy parameter ε. We also provide an efficient numerical method to solve the convex max-min program that defines the characteristic time. Our method is based on a complete characterization of the alternative bandit instances that the optimal sampling strategy needs to rule out, thus making our bound tighter than the one provided by <cit.>. Using this method, we propose a Track-and-Stop algorithm that identifies the set of ε-good arms w.h.p and enjoys asymptotic optimality (when δ goes to zero) in terms of the expected sample complexity. Finally, using numerical simulations, we demonstrate our algorithm's advantage over state-of-the-art methods, even for moderate values of the risk parameter.
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