We give a simple and computationally efficient algorithm that, for any constant $\varepsilon>0$, obtains $\varepsilon T$-swap regret within only $T = \mathsf{polylog}(n)$ rounds; this is an exponential improvement compared to the super-linear number of rounds required by the state-of-the-art algorithm, and resolves the main open problem of [Blum and Mansour 2007]. Our algorithm has an exponential dependence on $\varepsilon$, but we prove a new, matching lower bound. Our algorithm for swap regret implies faster convergence to $\varepsilon$-Correlated Equilibrium ($\varepsilon$-CE) in several regimes: For normal form two-player games with $n$ actions, it implies the first uncoupled dynamics that converges to the set of $\varepsilon$-CE in polylogarithmic rounds; a $\mathsf{polylog}(n)$-bit communication protocol for $\varepsilon$-CE in two-player games (resolving an open problem mentioned by [Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]; and an $\tilde{O}(n)$-query algorithm for $\varepsilon$-CE (resolving an open problem of [Babichenko'2020] and obtaining the first separation between $\varepsilon$-CE and $\varepsilon$-Nash equilibrium in the query complexity model). For extensive-form games, our algorithm implies a PTAS for $\mathit{normal}$ $\mathit{form}$ $\mathit{correlated}$ $\mathit{equilibria}$, a solution concept often conjectured to be computationally intractable (e.g. [Stengel-Forges'08, Fujii'23]).

翻译：我们提出一种简单且计算高效的算法，对于任意常数 $\varepsilon>0$，仅需 $T = \mathsf{polylog}(n)$ 轮即可获得 $\varepsilon T$-交换遗憾；这相较于现有最优算法所需的超线性轮数实现了指数级改进，并解决了[Blum and Mansour 2007]中的主要开放问题。我们的算法对 $\varepsilon$ 具有指数依赖性，但我们证明了新的、匹配的下界。我们的交换遗憾算法在多个场景中实现了对 $\varepsilon$-相关均衡 ($\varepsilon$-CE) 的更快收敛：对于具有 $n$ 个动作的规范形式双人博弈，它首次实现了在多对数轮内收敛到 $\varepsilon$-CE 集合的非耦合动态；一种用于双人博弈中 $\varepsilon$-CE 的 $\mathsf{polylog}(n)$ 比特通信协议（解决了[Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]提及的开放问题）；以及一种用于 $\varepsilon$-CE 的 $\tilde{O}(n)$ 查询算法（解决了[Babichenko'2020]的开放问题，并在查询复杂度模型中首次实现了 $\varepsilon$-CE 与 $\varepsilon$-纳什均衡之间的分离）。对于扩展形式博弈，我们的算法为 $\mathit{规范形式相关均衡}$（一种常被推测为计算棘手的解概念，例如[Stengel-Forges'08, Fujii'23]）提供了 PTAS。