  We study the combinatorial pure exploration problem Best-Set in stochastic
multi-armed bandits. In a Best-Set instance, we are given $n$ arms with unknown
reward distributions, as well as a family $\mathcal{F}$ of feasible subsets
over the arms. Our goal is to identify the feasible subset in $\mathcal{F}$
with the maximum total mean using as few samples as possible. The problem
generalizes the classical best arm identification problem and the top-$k$ arm
identification problem, both of which have attracted significant attention in
recent years. We provide a novel instance-wise lower bound for the sample
complexity of the problem, as well as a nontrivial sampling algorithm, matching
the lower bound up to a factor of $\ln|\mathcal{F}|$. For an important class of
combinatorial families, we also provide polynomial time implementation of the
sampling algorithm, using the equivalence of separation and optimization for
convex program, and approximate Pareto curves in multi-objective optimization.
We also show that the $\ln|\mathcal{F}|$ factor is inevitable in general
through a nontrivial lower bound construction. Our results significantly
improve several previous results for several important combinatorial
constraints, and provide a tighter understanding of the general Best-Set
problem.
  We further introduce an even more general problem, formulated in geometric
terms. We are given $n$ Gaussian arms with unknown means and unit variance.
Consider the $n$-dimensional Euclidean space $\mathbb{R}^n$, and a collection
$\mathcal{O}$ of disjoint subsets. Our goal is to determine the subset in
$\mathcal{O}$ that contains the $n$-dimensional vector of the means. The
problem generalizes most pure exploration bandit problems studied in the
literature. We provide the first nearly optimal sample complexity upper and
lower bounds for the problem.
