Optimal Alternating Regret for Online Learning and Games

• URL: http://arxiv.org/abs/2608.24731v1
• Date: Tue, 25 Aug 2026 15:41:17 GMT
• Title: Optimal Alternating Regret for Online Learning and Games
• Abstract summary: We settle the minimax-optimal alternating regret, a regret notion motivated by alternating learning dynamics in games, for both online linear optimization (OLO) and online convex optimization (OCO)<n>For OLO over the probability simplex $_d$, we give an algorithm with $O(log d)$ alternating regret that remains a constant for any time horizon $T$, and a matching lower bound.<n>For general OCO over a $d$-dimensional compact convex set, we give an algorithm with $O(dlog (1+T/d)$ alternating regret, improving the previous best of
• Score: 13.57157968084561
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We settle the minimax-optimal alternating regret, a regret notion motivated by alternating learning dynamics in games, for both online linear optimization (OLO) and online convex optimization (OCO). For OLO over the probability simplex $Δ_d$, we give an algorithm with $O(\log d)$ alternating regret that remains a constant for any time horizon $T$, and a matching lower bound. Our constant regret bound significantly improves previous results with $O(\log ^{2/3}d \cdot T^{1/3})$ regret [Cevher, Cutkosky, Kavis, Piliouras, Skoulakis, Viano, NeurIPS 2023, Hait, Li, Luo, Zhang, COLT 2025]. As a result, we obtain alternating learning dynamics with $O(\log d /T)$ convergence to Nash equilibria in two-player zero-sum games and $O(\log d /T)$ convergence to coarse correlated equilibria in two-player general-sum games. This is the first uncoupled learning dynamics with $O(1/T)$ convergence to CCE in two-player general-sum games, while all prior works suffer additional $\log T$ factors. For general OCO over a $d$-dimensional compact convex set, we give an algorithm with $O(d\log (1+T/d))$ alternating regret, improving the previous best of $\widetilde{O}(d^{2/3}T^{1/3})$. We also prove a matching lower bound of $Ω(d\log (1+T/d))$, showing that the $Ω(\log T)$ factor is unavoidable.

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