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In this paper we establish efficient and \emph{uncoupled} learning dynamicsso that, when employed by all players in a general-sum multiplayer game, the\emph{swap regret} of each player after $T$ repetitions of the game is boundedby $O(\log T)$, improving over the prior best bounds of $O(\log^4 (T))$. At thesame time, we guarantee optimal $O(\sqrt{T})$ swap regret in the adversarialregime as well. To obtain these results, our primary contribution is to showthat when all players follow our dynamics with a \emph{time-invariant} learningrate, the \emph{second-order path lengths} of the dynamics up to time $T$ arebounded by $O(\log T)$, a fundamental property which could have furtherimplications beyond near-optimally bounding the (swap) regret. Our proposedlearning dynamics combine in a novel way \emph{optimistic} regularized learningwith the use of \emph{self-concordant barriers}. Further, our analysis isremarkably simple, bypassing the cumbersome framework of higher-ordersmoothness recently developed by Daskalakis, Fishelson, and Golowich(NeurIPS'21).

Uncoupled Learning Dynamics with $O(\log T)$ Swap Regret in Multiplayer Games