  Combinatorial bandits with semi-bandit feedback generalize multi-armed
bandits, where the agent chooses sets of arms and observes a noisy reward for
each arm contained in the chosen set. The action set satisfies a given
structure such as forming a base of a matroid or a path in a graph. We focus on
the pure-exploration problem of identifying the best arm with fixed confidence,
as well as a more general setting, where the structure of the answer set
differs from the one of the action set. Using the recently popularized game
framework, we interpret this problem as a sequential zero-sum game and develop
a CombGame meta-algorithm whose instances are asymptotically optimal algorithms
with finite time guarantees. In addition to comparing two families of learners
to instantiate our meta-algorithm, the main contribution of our work is a
specific oracle efficient instance for best-arm identification with
combinatorial actions. Based on a projection-free online learning algorithm for
convex polytopes, it is the first computationally efficient algorithm which is
asymptotically optimal and has competitive empirical performance.
