Fast Convergence of Regularized Learning in Games

We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at O(T {-3/4}), while the sum of utilities converges to an approximate optimum at O(T {-1}) --an improvement upon the worst case O(T {-1/2}) rates. We show a black-box reduction for any algorithm in the class to achieve \tilde{O}(T {-1/2}) rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of Rakhlin and Shridharan \cite{Rakhlin2013} and Daskalakis et al. \cite{Daskalakis2014}, who only analyzed two-player zero-sum games for specific algorithms.