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：我们提出了一种简单且计算高效的算法，对于任意常数$\varepsilon>0$，该算法仅需$T = \mathsf{polylog}(n)$轮即可获得$\varepsilon T$-交换遗憾；相比现有最优算法所需的超线性轮数，这是一个指数级改进，并解决了[Blum and Mansour 2007]的主要公开问题。我们的算法对$\varepsilon$呈指数依赖，但我们也证明了新的匹配下界。该交换遗憾算法在若干场景下实现了对$\varepsilon$-相关均衡（$\varepsilon$-CE）的更快收敛：对于包含$n$个动作的二人正规型博弈，它首次实现了在多项式对数轮内收敛到$\varepsilon$-CE集合的非耦合动态；提出了一个用于二人博弈中$\varepsilon$-CE的$\mathsf{polylog}(n)$比特通信协议（解决了[Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]提及的公开问题）；以及一个用于$\varepsilon$-CE的$\tilde{O}(n)$查询算法（解决了[Babichenko'2020]的公开问题，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）。对于扩展型博弈，我们的算法为$\mathit{正规型相关均衡}$（一种常被推测为计算难解的解概念，例如[Stengel-Forges'08, Fujii'23]）提供了多项式时间近似方案（PTAS）。