Selecting the best system and multi-armed bandits

Consider the problem of finding a population or a probability distribution amongst many with the largest mean when these means are unknown but population samples can be simulated or otherwise generated. Typically, by selecting largest sample mean population, it can be shown that false selection probability decays at an exponential rate. Lately, researchers have sought algorithms that guarantee that this probability is restricted to a small $\delta$ in order $\log(1/\delta)$ computational time by estimating the associated large deviations rate function via simulation. We show that such guarantees are misleading when populations have unbounded support even when these may be light-tailed. Specifically, we show that any policy that identifies the correct population with probability at least $1-\delta$ for each problem instance requires infinite number of samples in expectation in making such a determination in any problem instance. This suggests that some restrictions are essential on populations to devise $O(\log(1/\delta))$ algorithms with $1 - \delta$ correctness guarantees. We note that under restriction on population moments, such methods are easily designed, and that sequential methods from stochastic multi-armed bandit literature can be adapted to devise such algorithms.