Optimal Alternating Regret for Online Learning and Games

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 $Θ(\log d)$, independent of the horizon $T$. This significantly improves upon the best-known $\mathcal{O}(T^{1/3}\log^{2/3} d)$ established by Hait et al. [2025]. 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 $Θ\left(d\log \left(1+\frac{T}{d}\right)\right)$, also significantly improving upon the best-known $\mathcal{O}((d\log T)^{2/3}T^{1/3})$ upper bound and resolving the open problem posed by Cevher et al. [2023], Hait et al. [2025]. Technically, our upper bound for the expert problem is achieved by a corrected variant of Hedge, in which carefully designed correction terms cancel the unfavorable curvature arising in the alternating-regret analysis. 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 expert construction 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.

We introduce an uncoupled learning algorithm which, when employed by all players of an arbitrary $N$-player normal form game with up to $K$ actions per player, guarantees $O(N^3\log^2 K)$ individual regret, uniformly over the horizon of play. The proposed algorithm - which we call higher-order optimism with discounting (HOOD) is a variant of optimistic follow-the-regularized-leader (OptFTRL) that combines a discounted $(N+1)$-th order predictor with entropic regularization over a suitable "lifting" of the game's strategy space. This combination of ingredients is purposefully designed to dampen large oscillations of the induced sequence of play in a controlled manner, removing in this way a key stumbling block of previous attempts to achieve constant regret in general games. Our approach bears several striking similarities to the concurrent - and completely independent - work of Liu, Farina, and Ozdaglar (arXiv:2608.31166), who very recently derived an $O(N^{21}\log^{4} K)$ regret bound through the use of higher-order optimism and an exponential moving average estimator.

Decentralized Online Convex Optimization (D-OCO) is a fundamental framework in which a set of local agents aims to minimize a sequence of global loss functions using only local computations and communications. While recent work has established near-optimal regret guarantees for convex and strongly convex losses, the case of exp-concave losses remains an open challenge: specifically, whether logarithmic regret is achievable. In this paper, we answer this question affirmatively by proposing Decentralized Online Newton Step (D-ONS), which adapts the idea of blocking and accelerated gossip to the Online Newton Step algorithm in a non-trivial manner. Concretely, our algorithm achieves a regret bound of $O\left(Nd(1-σ_2)^{-1/2}\,\log^2(T) \right)$ where $N$ is the number of agents, $d$ is the dimension, $T$ is the horizon, and $1-σ_2$ represents the spectral gap of the network. To our knowledge, this is the first logarithmic regret bound for decentralized online exp-concave optimization. We further establish a nearly matching lower bound of $Ω\left(Nd (1-σ_2)^{-1/2}\,\log(T/N) \right)$ showing the near optimality of our upper bound. Finally, we extend our algorithm to decentralized online linear regression in which the feasible domain is unbounded and features and labels are locally observed. By gossiping covariance and label-feature statistics, we also achieve logarithmic regret.