Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Self-play via online learning is one of the premier ways to solve large-scale twoplayer 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 Õ(1/T) convergence to coarse correlated equilibria even in generalsum 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 (1/ 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 δ > 0, there exists a 2 2 matrix game such that the algorithm admits a constant duality gap even after 1/δ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.