Hedging in games: Faster convergence of external and swap regrets
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We consider the setting where players run the Hedge algorithm or its optimistic variant <cit.> 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 Õ( 1/T ^5/6 ), improving the previous bound O(1/T^3/4) by <cit.>. 2) In contrast, we show that the convergence rate of vanilla Hedge is no better than Ω̃(1/ √(T)), addressing an open question posted in <cit.>. For general m-player games, we show that the swap regret of each player decays at rate Õ(m^1/2 (n/T)^3/4) when they combine optimistic Hedge with the classical external-to-internal reduction of Blum and Mansour <cit.>. The algorithm can also be modified to achieve the same rate against itself and a rate of Õ(√(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.
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