Disagreement-based combinatorial pure exploration: Efficient algorithms and an analysis with localization

We design new algorithms for the combinatorial pure exploration problem in the multi-arm bandit framework. In this problem, we are given K distributions and a collection of subsets $\mathcal{V} \subset 2^K$ of these distributions, and we would like to find the subset $v \in \mathcal{V}$ that has largest cumulative mean, while collecting, in a sequential fashion, as few samples from the distributions as possible. We study both the fixed budget and fixed confidence settings, and our algorithms essentially achieve state-of-the-art performance in all settings, improving on previous guarantees for structures like matchings and submatrices that have large augmenting sets. Moreover, our algorithms can be implemented efficiently whenever the decision set V admits linear optimization. Our analysis involves precise concentration-of-measure arguments and a new algorithm for linear programming with exponentially many constraints.