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