We study the problem of selecting a subset from a large action space shared by a family of bandits. In many natural situations, while the nominal set of actions is large, actions are highly correlated: many yield similar rewards across environments, making it wasteful to maintain the full set. Our aim is to understand whether it is possible -- and how -- to select a smaller set of representative actions that performs nearly as well as the full action space. Our main contribution is a surprisingly simple algorithm: repeatedly sample a bandit instance at random, solve it, and collect the optimal action. This algorithm can significantly reduce the action space when such correlations are present, without the need to know a-priori the correlation structure. We provide theoretical guarantees on the performance of the algorithm and demonstrate its practical effectiveness through empirical comparisons with Combinatorial Bandit, Meta Learning Bandit and Zooming baselines.

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