Near-optimal no-regret learning for correlated equilibria in multi-player general-sum games

Recently, Daskalakis, Fishelson, and Golowich ([DFG21] 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(polylog(T )) after T repetitions of the game. In this paper we extend their result from external regret to internal and swap regret, thereby establishing uncoupled learning dynamics that converge to an approximate correlated equilibrium at the rate of 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 ([CP20] NeurIPS '20), and it is optimal 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 first establish that the nointernal-regret learning dynamics of Stoltz and Lugosi ([SL05] 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 [DFG21] to near-optimally bound the internal regret.

Moreover, we establish an O(polylog(T )) no-swap-regret bound for the classic algorithm of Blum and Mansour ([BM07] JMLR '07). We do so by introducing a technique based on the Cauchy Integral Formula that circumvents the more limited combinatorial arguments of [DFG21]. In addition to shedding clarity on the near-optimal regret guarantees of [BM07], our arguments provide insights into the various ways in which the techniques by [DFG21] can be extended and leveraged in the analysis of more involved learning algorithms.