Minimax Alternating Regret for the Experts Problem and Online Convex Optimization

arXiv:2608.25182v2 Announce Type: replace-cross Abstract: In this paper, we study alternating regret in online convex optimization (OCO), motivated by the success of alternating learning dynamics in two-player games. Although previous works have shown that $o(\sqrt{T})$ alternating regret is achievable under various assumptions on the loss functions and feasible domains, the minimax regret rate has remained open even for the expert problem. In this paper, we resolve this question by showing matching lower and upper bounds for both the expert problem and general OCO. Somewhat surprisingly, for the $d$-expert problem, we show that the minimax alternating regret is $\Theta(\log d)$ by providing both a lower bound and an algorithm achieving a matching upper bound. This significantly improves upon the $O(T^{1/3}\log^{2/3} d)$ established by [HLLZ25]. We further extend our results to general OCO over a $d$-dimensional compact convex set and prove that the worst-case minimax alternating regret is $\Theta\left(d\log \left(1+\frac{T}{d}\right)\right)$, also significantly improving upon the $O((d\log T)^{2/3}T^{1/3})$ upper bound and resolving the open problem posed by [CCK+23,HLLZ25]. As a direct implication, our alternating dynamics achieve $O(\frac{\log d}{T})$ convergence to Nash equilibrium in two-player zero-sum games, matching the state of the art, and give the first such convergence rate to coarse correlated equilibrium in two-player general-sum games. Technically, our upper bound for the expert problem is achieved by a corrected variant of Hedge. We extend the same corrected-potential argument to continuous action sets to obtain the optimal alternating-regret rate for OCO. For the lower bounds, the construction for the expert problem repeatedly eliminates half of the candidate experts, while the OCO lower bound instance construction replaces this discrete elimination by a more involved multiscale construction on the unit disk.