Finding All $\epsilon$-Good Arms in Stochastic Bandits

The pure-exploration problem in stochastic multi-armed bandits aims to find one or more arms with the largest (or near largest) means. Examples include finding an ✏-good arm, best-arm identification, top-k arm identification, and finding all arms with means above a specified threshold. However, the problem of finding all ✏-good arms has been overlooked in past work, although arguably this may be the most natural objective in many applications. For example, a virologist may conduct preliminary laboratory experiments on a large candidate set of treatments and move all ✏-good treatments into more expensive clinical trials. Since the ultimate clinical efficacy is uncertain, it is important to identify all ✏-good candidates. Mathematically, the all-✏-good arm identification problem presents significant new challenges and surprises that do not arise in the pure-exploration objectives studied in the past. We introduce two algorithms to overcome these and demonstrate their great empirical performance on a large-scale crowd-sourced dataset of 2.2M ratings collected by the New Yorker Caption Contest as well as a dataset testing hundreds of possible cancer drugs.