  Self-play via online learning is one of the premier ways to solve large-scale
two-player zero-sum games, both in theory and practice. Particularly popular
algorithms include optimistic multiplicative weights update (OMWU) and
optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$
ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU
offers several advantages including logarithmic dependence on the size of the
payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated
equilibria even in general-sum games. However, in terms of last-iterate
convergence in two-player zero-sum games, an increasingly popular topic in this
area, OGDA guarantees that the duality gap shrinks at a rate of
$O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU
depends on some game-dependent constant that could be arbitrarily large. This
begs the question: is this potentially slow last-iterate convergence an
inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat
surprisingly, we show that the former is true. More generally, we prove that a
broad class of algorithms that do not forget the past quickly all suffer the
same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$
matrix game such that the algorithm admits a constant duality gap even after
$1/\delta$ rounds. This class of algorithms includes OMWU and other standard
optimistic follow-the-regularized-leader algorithms.
