Stochastic No-regret Learning for General Games with Variance Reduction

We show that a stochastic version of optimistic mirror descent (OMD), a variant of mirror descent with recency bias, converges fast in general games. More specifically, with our algorithm, the individual regret of each player vanishes at a speed of $O(1/T^{3/4})$ and the sum of all players' regret vanishes at a speed of $O(1/T)$, which is an improvement upon the $O(1/\sqrt{T})$ convergence rate of prior stochastic algorithms, where $T$ is the number of interaction rounds. Due to the advantage of stochastic methods in the computational cost, we significantly improve the time complexity over the deterministic algorithms to approximate coarse correlated equilibrium. To achieve lower time complexity, we equip the stochastic version of OMD in \cite{alacaoglu2021stochastic} with a novel low-variance Monte-Carlo estimator. Our algorithm extends previous works \cite{alacaoglu2021stochastic,carmon2019variance} from two-player zero-sum games to general games.