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An Optimal Elimination Algorithm for Learning a Best Arm

We consider the classic problem of $(ε,δ)$-PAC learning a best arm where the goal is to identify with confidence $1-δ$ an arm whose mean is an $ε$-approximation to that of the highest mean arm in a multi-armed bandit setting. This problem is one of the most fundamental problems in statistics and learning theory, yet somewhat surprisingly its worst-case sample complexity is not well understood. In this paper, we propose a new approach for $(ε,δ)$-PAC learning a best arm. This approach leads to an algorithm whose sample complexity converges to \emph{exactly} the optimal sample complexity of $(ε,δ)$-learning the mean of $n$ arms separately and we complement this result with a conditional matching lower bound. More specifically: