For distributions with a known bound on an increasing convex moment, the AL1 algorithm identifies the best arm with probability at least 1-δ using asymptotically minimal expected samples as δ goes to 0.
Lemma 28 For any ua∈ℜ , non-negative constants ˜Ba and rectangle Ga, P ( K⋂ a=1 { max λa∈Ga L(λa, m(µa), ˆµa(n))≥ ua,Cd a }) ≤ K ∏ a=1 e ˜Bae−taua
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Optimal $\delta$-Correct Best-Arm Selection for Heavy-Tailed Distributions
For distributions with a known bound on an increasing convex moment, the AL1 algorithm identifies the best arm with probability at least 1-δ using asymptotically minimal expected samples as δ goes to 0.