  Motivated by applications in Game Theory, Optimization, and Generative
Adversarial Networks, recent work of Daskalakis et al \cite{DISZ17} and
follow-up work of Liang and Stokes \cite{LiangS18} have established that a
variant of the widely used Gradient Descent/Ascent procedure, called
"Optimistic Gradient Descent/Ascent (OGDA)", exhibits last-iterate convergence
to saddle points in {\em unconstrained} convex-concave min-max optimization
problems. We show that the same holds true in the more general problem of {\em
constrained} min-max optimization under a variant of the no-regret
Multiplicative-Weights-Update method called "Optimistic Multiplicative-Weights
Update (OMWU)". This answers an open question of Syrgkanis et al \cite{SALS15}.
  The proof of our result requires fundamentally different techniques from
those that exist in no-regret learning literature and the aforementioned
papers. We show that OMWU monotonically improves the Kullback-Leibler
divergence of the current iterate to the (appropriately normalized) min-max
solution until it enters a neighborhood of the solution. Inside that
neighborhood we show that OMWU is locally (asymptotically) stable converging to
the exact solution. We believe that our techniques will be useful in the
analysis of the last iterate of other learning algorithms.

Optimization and Control;Computer Science and Game Theory;Machine Learning