  We show that Optimistic Hedge -- a common variant of
multiplicative-weights-updates with recency bias -- attains ${\rm poly}(\log
T)$ regret in multi-player general-sum games. In particular, when every player
of the game uses Optimistic Hedge to iteratively update her strategy in
response to the history of play so far, then after $T$ rounds of interaction,
each player experiences total regret that is ${\rm poly}(\log T)$. Our bound
improves, exponentially, the $O({T}^{1/2})$ regret attainable by standard
no-regret learners in games, the $O(T^{1/4})$ regret attainable by no-regret
learners with recency bias (Syrgkanis et al., 2015), and the ${O}(T^{1/6})$
bound that was recently shown for Optimistic Hedge in the special case of
two-player games (Chen & Pen, 2020). A corollary of our bound is that
Optimistic Hedge converges to coarse correlated equilibrium in general games at
a rate of $\tilde{O}\left(\frac 1T\right)$.
