  A matroid is a notion of independence in combinatorial optimization which is
closely related to computational efficiency. In particular, it is well known
that the maximum of a constrained modular function can be found greedily if and
only if the constraints are associated with a matroid. In this paper, we bring
together the ideas of bandits and matroids, and propose a new class of
combinatorial bandits, matroid bandits. The objective in these problems is to
learn how to maximize a modular function on a matroid. This function is
stochastic and initially unknown. We propose a practical algorithm for solving
our problem, Optimistic Matroid Maximization (OMM); and prove two upper bounds,
gap-dependent and gap-free, on its regret. Both bounds are sublinear in time
and at most linear in all other quantities of interest. The gap-dependent upper
bound is tight and we prove a matching lower bound on a partition matroid
bandit. Finally, we evaluate our method on three real-world problems and show
that it is practical.
