  In this work, we establish near-linear and strong convergence for a natural
first-order iterative algorithm that simulates Von Neumann's Alternating
Projections method in zero-sum games. First, we provide a precise analysis of
Optimistic Gradient Descent/Ascent (OGDA) -- an optimistic variant of Gradient
Descent/Ascent -- for \emph{unconstrained} bilinear games, extending and
strengthening prior results along several directions. Our characterization is
based on a closed-form solution we derive for the dynamics, while our results
also reveal several surprising properties. Indeed, our main algorithmic
contribution is founded on a geometric feature of OGDA we discovered; namely,
the limit points of the dynamics are the orthogonal projection of the initial
state to the space of attractors.
  Motivated by this property, we show that the equilibria for a natural class
of \emph{constrained} bilinear games are the intersection of the unconstrained
stationary points with the corresponding probability simplexes. Thus, we employ
OGDA to implement an Alternating Projections procedure, converging to an
$\epsilon$-approximate Nash equilibrium in
$\widetilde{\mathcal{O}}(\log^2(1/\epsilon))$ iterations. Our techniques
supplement the recent work in pursuing last-iterate guarantees in min-max
optimization. Finally, we illustrate an -- in principle -- trivial reduction
from any game to the assumed class of instances, without altering the space of
equilibria.
