We consider the setting where players run the Hedge algorithm or its optimistic variant \mbox{%DIFAUXCMD syrgkanis2015fast to play an n-action game repeatedly for T rounds. 1) For two-player games, we show that the regret of optimistic Hedge decays at O( 1/T ^{5/6} ), improving the previous bound O(1/T^{3/4}) by \mbox{%DIFAUXCMD syrgkanis2015fast 2) In contrast, we show that the convergence rate of vanilla Hedge is no better than \Omega(1/ T) , addressing an open question posted in \mbox{%DIFAUXCMD syrgkanis2015fast . For general m-player games, we show that the swap regret of each player decays at rate O(m^{1/2} (n/T)^{3/4}) when they combine optimistic Hedge with the classical external-to-internal reduction of Blum and Mansour \mbox{%DIFAUXCMD blum2007external . The algorithm can also be modified to achieve the same rate against itself and a rate of O(\sqrt{n/T}) against adversaries. Via standard connections, our upper bounds also imply faster convergence to coarse correlated equilibria in two-player games and to correlated equilibria in multiplayer games.
We consider the setting where players run the Hedge algorithm or its optimistic variant to play an n-action game repeatedly for T rounds. 1) For two-player games, we show that the regret of optimistic Hedge decays at O( 1/T ^{5/6} ), improving the previous bound O(1/T^{3/4}) by Syrgkanis, Agarwal, Luo and Schapire (NIPS'15) 2) In contrast, we show that the convergence rate of vanilla Hedge is no better than \Omega(1/ T) , addressing an open question posted in Syrgkanis, Agarwal, Luo and Schapire (NIPS'15) . For general m-player games, we show that the swap regret of each player decays at rate O(m^{1/2} (n/T)^{3/4}) when they combine optimistic Hedge with the classical external-to-internal reduction of Blum and Mansour (JMLR'07) . The algorithm can also be modified to achieve the same rate against itself and a rate of O(\sqrt{n/T}) against adversaries. Via standard connections, our upper bounds also imply faster convergence to coarse correlated equilibria in two-player games and to correlated equilibria in multiplayer games.
D
\mbox{%DIFAUXCMD syrgkanis2015fast
R
\mbox{%DIFAUXCMD syrgkanis2015fast
Syrgkanis, Agarwal, Luo and Schapire (NIPS'15)
R
\mbox{%DIFAUXCMD syrgkanis2015fast
Syrgkanis, Agarwal, Luo and Schapire (NIPS'15)
R
\mbox{%DIFAUXCMD blum2007external
(JMLR'07)