[Submitted on 4 Nov 2026]
Title:Optimistic Hedge in General-Sum Games: Fundamental Limits and Regret Bounds via Temporal Approximation
View a PDF of the paper titled Optimistic Hedge in General-Sum Games: Fundamental Limits and Regret Bounds via Temporal Approximation, by Yuval Dagan and 3 other authors
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Abstract:Optimistic Hedge, a multiplicative weight update with one step of extrapolation, is one of the most natural regret-minimizing dynamics for games, and characterizing its regret has been a longstanding open question. Daskalakis, Fishelson and Golowich proved that if all $m$ players in a finite general-sum game follow Optimistic Hedge with a fixed learning rate, each player's regret is polylogarithmic in the horizon $T$. Subsequently, Ha showed that an appropriate choice of learning rate makes each player's regret bounded independently of $T$. This bound, however, is non-constructive and its dependence on the number of actions $n$ is not explicit, leaving open whether Optimistic Hedge can achieve a constant regret bound which is polynomial in the size of the game.
We show that the answer hinges on the number of players. In two-player games, there is a learning rate for which Optimistic Hedge guarantees $O(n\log n)$ regret independent of the horizon. In three-player games, however, no such bound is possible: for any learning rate up to an absolute constant and for sufficiently large horizon $T$, there is a game with $n$ actions per player in which some player's regret is $2^{\Omega(n)}$, and more generally, for every $m\ge3$ we construct $m$-player games in which some player's regret is $2^{\Omega(mn)}$; for learning rates above $2^{-\Omega(mn)}$, that player's regret even grows linearly in $T$.
Our upper bound is based on a temporal approximation framework for the analysis of Optimistic Hedge. Starting from the variance-based RVU bound of Daskalakis et al., we approximate the losses over time-localized windows by well-behaved functions, whose total variation is a small fraction of their cumulative variance. In $m$-player games, the same framework gives a significantly simplified proof of the recent $O(\sqrt{m}\log n\log T)$ regret bound due to Ha.
Subjects:
Machine Learning (cs.LG); Computer Science and Game Theory (cs.GT)
Cite as: arXiv:2611.07078 [cs.LG]
(or arXiv:2611.07078v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2611.07078
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Yuval Dagan [view email] [v1] Wed, 4 Nov 2026 19:34:53 UTC (65 KB)
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