  In this paper we establish efficient and \emph{uncoupled} learning dynamics
so that, when employed by all players in a general-sum multiplayer game, the
\emph{swap regret} of each player after $T$ repetitions of the game is bounded
by $O(\log T)$, improving over the prior best bounds of $O(\log^4 (T))$. At the
same time, we guarantee optimal $O(\sqrt{T})$ swap regret in the adversarial
regime as well. To obtain these results, our primary contribution is to show
that when all players follow our dynamics with a \emph{time-invariant} learning
rate, the \emph{second-order path lengths} of the dynamics up to time $T$ are
bounded by $O(\log T)$, a fundamental property which could have further
implications beyond near-optimally bounding the (swap) regret. Our proposed
learning dynamics combine in a novel way \emph{optimistic} regularized learning
with the use of \emph{self-concordant barriers}. Further, our analysis is
remarkably simple, bypassing the cumbersome framework of higher-order
smoothness recently developed by Daskalakis, Fishelson, and Golowich
(NeurIPS'21).
