Uncoupled Learning Dynamics with O(log T) Swap Regret in Multiplayer Games
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In this paper we establish efficient and uncoupled learning dynamics so that, when employed by all players in a general-sum multiplayer game, the swap regret of each player after T repetitions of the game is bounded by O(log T), improving over the prior best bounds of O(log^4 (T)). At the same time, we guarantee optimal O(√(T)) swap regret in the adversarial regime as well. To obtain these results, our primary contribution is to show that when all players follow our dynamics with a time-invariant learning rate, the second-order path lengths of the dynamics up to time T are bounded by O(log T), a fundamental property which could have further implications beyond near-optimally bounding the (swap) regret. Our proposed learning dynamics combine in a novel way optimistic regularized learning with the use of self-concordant barriers. Further, our analysis is remarkably simple, bypassing the cumbersome framework of higher-order smoothness recently developed by Daskalakis, Fishelson, and Golowich (NeurIPS'21).
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