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

### Ioannis Anagnostides · Gabriele Farina · Christian Kroer · Chung-Wei Lee · Haipeng Luo · Tuomas Sandholm

##### Hall J #839

Keywords: [ Uncoupled learning dynamics ] [ optimism ] [ swap regret ] [ correlated equilibria ]

[ Abstract ]
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Thu 1 Dec 9 a.m. PST — 11 a.m. PST

Abstract: In this paper we establish efficient and \emph{uncoupled} learning dynamics so 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 bounded by $O(\log T)$, improving over the prior best bounds of $O(\log^4 (T))$. At the same time, we guarantee optimal $O(\sqrt{T})$ 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 \emph{time-invariant} learning rate, the \emph{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 \emph{optimistic} regularized learning with the use of \emph{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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