  In order to find Nash-equilibria for two-player zero-sum games where each
player plays combinatorial objects like spanning trees, matchings etc, we
consider two online learning algorithms: the online mirror descent (OMD)
algorithm and the multiplicative weights update (MWU) algorithm. The OMD
algorithm requires the computation of a certain Bregman projection, that has
closed form solutions for simple convex sets like the Euclidean ball or the
simplex. However, for general polyhedra one often needs to exploit the general
machinery of convex optimization. We give a novel primal-style algorithm for
computing Bregman projections on the base polytopes of polymatroids. Next, in
the case of the MWU algorithm, although it scales logarithmically in the number
of pure strategies or experts $N$ in terms of regret, the algorithm takes time
polynomial in $N$; this especially becomes a problem when learning
combinatorial objects. We give a general recipe to simulate the multiplicative
weights update algorithm in time polynomial in their natural dimension. This is
useful whenever there exists a polynomial time generalized counting oracle
(even if approximate) over these objects. Finally, using the combinatorial
structure of symmetric Nash-equilibria (SNE) when both players play bases of
matroids, we show that these can be found with a single projection or convex
minimization (without using online learning).
