  This paper studies a multi-armed bandit (MAB) version of the range-searching
problem. In its basic form, range searching considers as input a set of points
(on the real line) and a collection of (real) intervals. Here, with each
specified point, we have an associated weight, and the problem objective is to
find a maximum-weight point within every given interval.
  The current work addresses range searching with stochastic weights: each
point corresponds to an arm (that admits sample access) and the point's weight
is the (unknown) mean of the underlying distribution. In this MAB setup, we
develop sample-efficient algorithms that find, with high probability,
near-optimal arms within the given intervals, i.e., we obtain PAC (probably
approximately correct) guarantees. We also provide an algorithm for a
generalization wherein the weight of each point is a multi-dimensional vector.
The sample complexities of our algorithms depend, in particular, on the size of
the optimal hitting set of the given intervals.
  Finally, we establish lower bounds proving that the obtained sample
complexities are essentially tight. Our results highlight the significance of
geometric constructs -- specifically, hitting sets -- in our MAB setting.
