Optimistic Mirror Descent Either Converges to Nash or to Strong Coarse Correlated Equilibria in Bimatrix Games

Abstract

We show that, for any sufficiently small fixed $\epsilon > 0$, when both players in a general-sum two-player (bimatrix) game employ optimistic mirror descent (OMD) with smooth regularization, learning rate $\eta = O(\epsilon^2)$ and $T = \Omega(\text{poly}(1/\epsilon))$ repetitions, either the dynamics reach an $\epsilon$-approximate Nash equilibrium (NE), or the average correlated distribution of play is an $\Omega(\text{poly}(\epsilon))$-strong coarse correlated equilibrium (CCE): any possible unilateral deviation does not only leave the player worse, but will decrease its utility by $\Omega(\text{poly}(\epsilon))$. As an immediate consequence, when the iterates of OMD are bounded away from being Nash equilibria in a bimatrix game, we guarantee convergence to an exact CCE after only $O(1)$ iterations. Our results reveal that uncoupled no-regret learning algorithms can converge to CCE in general-sum games remarkably faster than to NE in, for example, zero-sum games. To establish this, we show that when OMD does not reach arbitrarily close to a NE, the (cumulative) regret of both players is not only negative, but decays linearly with time. Given that regret is the canonical measure of performance in online learning, our results suggest that cycling behavior of no-regret learning algorithms in games can be justified in terms of efficiency.