A Logarithmic Regret Bound for Optimistic Hedge in General-Sum Games

Can simple no-regret dynamics attain smaller regret in self-play than against arbitrary adversaries? In $n$-player general-sum games, Daskalakis et al. 2021 proved an $O(n\log d_i{\log }^{4}T)$ individual regret bound for Optimistic Hedge, which improves upon the classical $O(\sqrt{T})$ adversarial regret bound. In this work, we show that Optimistic Hedge with a constant step size can further achieve $O(\sqrt{n}\log d_i\log T)$ individual external regret under expected loss-vector feedback. The time-averaged play consequently enjoys a coarse correlated equilibrium gap $O(\sqrt{n}\log d\log T/T)$, where $d={\max }_i{d}_i$. The improvement comes from a larger admissible step size $\eta =\Theta (1/(\sqrt{n}\log T))$. Our analysis proves factorial bounds on high-order differences of probability-weighted pairwise loss gaps, then applies finite-difference interpolation in a fixed Euclidean norm. These estimates sharpen the analysis of Daskalakis et al. 2021 and yield a logarithmic regret bound.