The True Sample Complexity of Identifying Good Arms

We consider two multi-armed bandit problems with $n$ arms: (i) given an $\epsilon > 0$, identify an arm with mean that is within $\epsilon$ of the largest mean and (ii) given a threshold $\mu0$ and integer $k$, identify $k$ arms with means larger than $\mu0$. Existing lower bounds and algorithms for the PAC framework suggest that both of these problems require $\Omega(n)$ samples. However, we argue that these definitions not only conflict with how these algorithms are used in practice, but also that these results disagree with intuition that says (i) requires only $\Theta(\frac{n}{m})$ samples where $m = |{ i : \mui > \max{i \in [n]} \mui - \epsilon}|$ and (ii) requires $\Theta(\frac{n}{m}k)$ samples where $m = |{ i : \mui > \mu_0 }|$. We provide definitions that formalize these intuitions, obtain lower bounds that match the above sample complexities, and develop explicit, practical algorithms that achieve nearly matching upper bounds.