  In this paper, we consider a bandit problem in which there are a number of
groups each consisting of infinitely many arms. Whenever a new arm is requested
from a given group, its mean reward is drawn from an unknown reservoir
distribution (different for each group), and the uncertainty in the arm's mean
reward can only be reduced via subsequent pulls of the arm. The goal is to
identify the infinite-arm group whose reservoir distribution has the highest
$(1-\alpha)$-quantile (e.g., median if $\alpha = \frac{1}{2}$), using as few
total arm pulls as possible. We introduce a two-step algorithm that first
requests a fixed number of arms from each group and then runs a finite-arm
grouped max-quantile bandit algorithm. We characterize both the
instance-dependent and worst-case regret, and provide a matching lower bound
for the latter, while discussing various strengths, weaknesses, algorithmic
improvements, and potential lower bounds associated with our instance-dependent
upper bounds.
