  We design new algorithms for the combinatorial pure exploration problem in
the multi-arm bandit framework. In this problem, we are given $K$ distributions
and a collection of subsets $\mathcal{V} \subset 2^{[K]}$ of these
distributions, and we would like to find the subset $v \in \mathcal{V}$ that
has largest mean, while collecting, in a sequential fashion, as few samples
from the distributions as possible. In both the fixed budget and fixed
confidence settings, our algorithms achieve new sample-complexity bounds that
provide polynomial improvements on previous results in some settings. Via an
information-theoretic lower bound, we show that no approach based on uniform
sampling can improve on ours in any regime, yielding the first interactive
algorithms for this problem with this basic property. Computationally, we show
how to efficiently implement our fixed confidence algorithm whenever
$\mathcal{V}$ supports efficient linear optimization. Our results involve
precise concentration-of-measure arguments and a new algorithm for linear
programming with exponentially many constraints.
