Pure Exploration with Infinite Answers

We study pure exploration problems in which the set of correct answers is possibly infinite. For example, such problems arise when regressing a continuous function on the means of the bandit or when learning Nash equilibria by querying noisy values of the payoff matrix. We derive an instance-dependent lower bound for these problems. By analyzing it, we discuss why existing methods (i.e., Sticky Track-and-Stop) for finite answer problems fail at being asymptotically optimal in this more general setting. Finally, we present a framework, Sticky-Sequence Track-and-Stop, which generalizes both Track-and-Stop and Sticky Track-and-Stop, and that enjoys asymptotic optimality. Due to its generality, our analysis also highlights special cases where existing methods enjoy optimality.