  We propose a simple variant of the generalized Frank-Wolfe method for solving
strongly convex composite optimization problems, by introducing an additional
averaging step on the dual variables. We show that in this variant, one can
choose a simple constant step-size and obtain a linear convergence rate on the
duality gaps. By leveraging the convergence analysis of this variant, we then
analyze the local convergence rate of the logistic fictitious play algorithm,
which is well-established in game theory but lacks any form of convergence rate
guarantees. We show that, with high probability, this algorithm converges
locally at rate $O(1/t)$, in terms of certain expected duality gap.
