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Learning the minimum/maximum mean among a finite set of distributions is afundamental sub-task in planning, game tree search and reinforcement learning.We formalize this learning task as the problem of sequentially testing how theminimum mean among a finite set of distributions compares to a given threshold.We develop refined non-asymptotic lower bounds, which show that optimalitymandates very different sampling behavior for a low vs high true minimum. Weshow that Thompson Sampling and the intuitive Lower Confidence Bounds policyeach nail only one of these cases. We develop a novel approach that we callMurphy Sampling. Even though it entertains exclusively low true minima, weprove that MS is optimal for both possibilities. We then design advancedself-normalized deviation inequalities, fueling more aggressive stopping rules.We complement our theoretical guarantees by experiments showing that MS worksbest in practice.

Sequential Test for the Lowest Mean: From Thompson to Murphy Sampling