  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 DFG 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.
