Recently, Daskalakis, Fishelson, and Golowich (DFG) (NeurIPS`21) showed that if all agents in a multi-player general-sum normal-form game employ Optimistic Multiplicative Weights Update (OMWU), the external regret of every player is $O(\textrm{polylog}(T))$ after $T$ repetitions of the game. We extend their result from external regret to internal regret and swap regret, thereby establishing uncoupled learning dynamics that converge to an approximate correlated equilibrium at the rate of $\tilde{O}(T^{-1})$. This substantially improves over the prior best rate of convergence for correlated equilibria of $O(T^{-3/4})$ due to Chen and Peng (NeurIPS`20), and it is optimal -- within the no-regret framework -- up to polylogarithmic factors in $T$. To obtain these results, we develop new techniques for establishing higher-order smoothness for learning dynamics involving fixed point operations. Specifically, we establish that the no-internal-regret learning dynamics of Stoltz and Lugosi (Mach Learn`05) are equivalently simulated by no-external-regret dynamics on a combinatorial space. This allows us to trade the computation of the stationary distribution on a polynomial-sized Markov chain for a (much more well-behaved) linear transformation on an exponential-sized set, enabling us to leverage similar techniques as DGF to near-optimally bound the internal regret. Moreover, we establish an $O(\textrm{polylog}(T))$ no-swap-regret bound for the classic algorithm of Blum and Mansour (BM) (JMLR`07). We do so by introducing a technique based on the Cauchy Integral Formula that circumvents the more limited combinatorial arguments of DFG. In addition to shedding clarity on the near-optimal regret guarantees of BM, our arguments provide insights into the various ways in which the techniques by DFG can be extended and leveraged in the analysis of more involved learning algorithms.

翻译：最近,Dakalakis、Fishelson和Golowich (DFG) (NeurIPS'21) 显示,如果多球员总和更普通的游戏中的所有代理商都采用最佳多倍增 WeWU, 每个玩家的外部遗憾是$O (textrm{polylogy}(T) 重复游戏之后的$T美元。我们将其外部遗憾的结果扩大到内部遗憾和互换遗憾,从而建立不相矛盾的学习链路程动态动态动态,以$tilde{O}(T ⁇ -1}) 的速变速率接近相对平衡。这大大改进了之前对QO(T ⁇ -3/4}) 和Peng (NeurIPS '20) 之前的最佳趋同比速度的趋同率。 在不重复游戏中,我们从不重复的框架中,可以提供多logyrical drivodition 因素。为了获得这些结果,我们开发了新技术来建立更高层次的平滑度, 学习固定点的动态, 接近Orental-ralal devodial dal devaldeal dismal disl) 。具体,我们通过学习一个不做一个不相像al deal dismal deal deal deal deal dismal deal deal deal disal disal disal disal 。