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We consider the setting where players run the Hedge algorithm or itsoptimistic variant to play an $n$-action game repeatedly for $T$ rounds.  1) For two-player games, we show that the regret of optimistic Hedge decaysat $\tilde{O}( 1/T ^{5/6} )$, improving the previous bound $O(1/T^{3/4})$ bySyrgkanis, Agarwal, Luo and Schapire (NIPS'15)  2) In contrast, we show that the convergence rate of vanilla Hedge is nobetter than $\tilde{\Omega}(1/ \sqrt{T})$, addressing an open question postedin Syrgkanis, Agarwal, Luo and Schapire (NIPS'15).  For general m-player games, we show that the swap regret of each playerdecays at rate $\tilde{O}(m^{1/2} (n/T)^{3/4})$ when they combine optimisticHedge with the classical external-to-internal reduction of Blum and Mansour(JMLR'07). The algorithm can also be modified to achieve the same rate againstitself and a rate of $\tilde{O}(\sqrt{n/T})$ against adversaries. Via standardconnections, our upper bounds also imply faster convergence to coarsecorrelated equilibria in two-player games and to correlated equilibria inmultiplayer games.

Hedging in games: Faster convergence of external and swap regrets