Sample complexity of partition identification using multi-armed bandits

Given a vector of probability distributions, or arms, each of which can be sampled independently, we consider the problem of identifying the partition to which this vector belongs from a finitely partitioned universe of such vector of distributions. We study this as a pure exploration problem in multi armed bandit settings and develop sample complexity bounds on the total mean number of samples required for identifying the correct partition with high probability. This framework subsumes well studied problems in the literature such as finding the best arm or the best few arms. We consider distributions belonging to the single parameter exponential family and primarily consider partitions where the vector of means of arms lie either in a given set or its complement. The sets considered correspond to distributions where there exists a mean above a specified threshold, where the set is a half space and where either the set or its complement is convex. When the set is convex we restrict our analysis to its complement being a union of half spaces. In all these settings, we characterize the lower bounds on mean number of samples for each arm. Further, inspired by the lower bounds, and building upon Garivier and Kaufmann 2016, we propose algorithms that can match these bounds asymptotically with decreasing probability of error. Applications of this framework may be diverse. We briefly discuss a few associated with finance.