An Optimal Elimination Algorithm for Learning a Best Arm

We consider the classic problem of (ǫ, δ)-PAC learning a best arm where the goal is to identify with confidence 1 -δ an arm whose mean is an ǫ-approximation to that of the highest mean arm in a multiarmed 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 (ǫ, δ)-PAC learning a best arm. This approach leads to an algorithm whose sample complexity converges to exactly the optimal sample complexity of (ǫ, δ)-learning the mean of n arms separately and we complement this result with a conditional matching lower bound. More specifically: • The algorithm's sample complexity converges to exactly n 2ǫ 2 log 1 δ as n grows and δ ≥ 1 n ; • We prove that no elimination algorithm obtains sample complexity arbitrarily lower than n 2ǫ 2 log 1 δ . Elimination algorithms is a broad class of (ǫ, δ)-PAC best arm learning algorithms that includes many algorithms in the literature. When n is independent of δ our approach yields an algorithm whose sample complexity converges to 2n ǫ 2 log 1 δ as n grows. In comparison with the best known algorithm for this problem our approach improves the sample complexity by a factor of over 1500 and over 6000 when δ ≥ 1 n .