  In this paper, we establish efficient and uncoupled learning dynamics so
that, when employed by all players in multiplayer perfect-recall
imperfect-information extensive-form games, the trigger regret of each player
grows as $O(\log T)$ after $T$ repetitions of play. This improves exponentially
over the prior best known trigger-regret bound of $O(T^{1/4})$, and settles a
recent open question by Bai et al. (2022). As an immediate consequence, we
guarantee convergence to the set of extensive-form correlated equilibria and
coarse correlated equilibria at a near-optimal rate of $\frac{\log T}{T}$.
  Building on prior work, at the heart of our construction lies a more general
result regarding fixed points deriving from rational functions with polynomial
degree, a property that we establish for the fixed points of (coarse) trigger
deviation functions. Moreover, our construction leverages a refined regret
circuit for the convex hull, which -- unlike prior guarantees -- preserves the
RVU property introduced by Syrgkanis et al. (NIPS, 2015); this observation has
an independent interest in establishing near-optimal regret under learning
dynamics based on a CFR-type decomposition of the regret.
