  We study the pure exploration problem subject to a matroid constraint
(Best-Basis) in a stochastic multi-armed bandit game. In a Best-Basis instance,
we are given $n$ stochastic arms with unknown reward distributions, as well as
a matroid $\mathcal{M}$ over the arms. Let the weight of an arm be the mean of
its reward distribution. Our goal is to identify a basis of $\mathcal{M}$ with
the maximum total weight, using as few samples as possible.
  The problem is a significant generalization of the best arm identification
problem and the top-$k$ arm identification problem, which have attracted
significant attentions in recent years. We study both the exact and PAC
versions of Best-Basis, and provide algorithms with nearly-optimal sample
complexities for these versions. Our results generalize and/or improve on
several previous results for the top-$k$ arm identification problem and the
combinatorial pure exploration problem when the combinatorial constraint is a
matroid.
