  Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative
Weights Update (OMWU) for saddle-point optimization have received growing
attention due to their favorable last-iterate convergence. However, their
behaviors for simple bilinear games over the probability simplex are still not
fully understood - previous analysis lacks explicit convergence rates, only
applies to an exponentially small learning rate, or requires additional
assumptions such as the uniqueness of the optimal solution. In this work, we
significantly expand the understanding of last-iterate convergence for OGDA and
OMWU in the constrained setting. Specifically, for OMWU in bilinear games over
the simplex, we show that when the equilibrium is unique, linear last-iterate
convergence is achieved with a learning rate whose value is set to a universal
constant, improving the result of (Daskalakis & Panageas, 2019b) under the same
assumption. We then significantly extend the results to more general objectives
and feasible sets for the projected OGDA algorithm, by introducing a sufficient
condition under which OGDA exhibits concrete last-iterate convergence rates
with a constant learning rate whose value only depends on the smoothness of the
objective function. We show that bilinear games over any polytope satisfy this
condition and OGDA converges exponentially fast even without the unique
equilibrium assumption. Our condition also holds for
strongly-convex-strongly-concave functions, recovering the result of (Hsieh et
al., 2019). Finally, we provide experimental results to further support our
theory.
