The Batch Complexity of Bandit Pure Exploration

In a fixed-confidence pure exploration problem in stochastic multi-armed bandits, an algorithm iteratively samples arms and should stop as early as possible and return the correct answer to a query about the arms distributions.We are interested in batched methods, which change their sampling behaviour only a few times, between batches of observations.We give an instance-dependent lower bound on the number of batches used by any sample efficient algorithm for any pure exploration task.We then give a general batched algorithm and prove upper bounds on its expected sample complexity and batch complexity.We illustrate both lower and upper bounds on best-arm identification and thresholding bandits.