  The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax
optimization problems, takes the descent and ascent steps either simultaneously
(Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to
converge faster, the performance gap between the two is not yet well understood
theoretically, especially in terms of global convergence rates. To address this
theory-practice gap, we present fine-grained convergence analyses of both
algorithms for strongly-convex-strongly-concave and Lipschitz-gradient
objectives. Our new iteration complexity upper bound of Alt-GDA is strictly
smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster.
Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general
algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main
idea is to alternately take gradients from extrapolations of the iterates. We
show that Alex-GDA satisfies a smaller iteration complexity bound, identical to
that of the Extra-gradient method, while requiring less gradient computations.
We also prove that Alex-GDA enjoys linear convergence for bilinear problems,
for which both Sim-GDA and Alt-GDA fail to converge at all.
