Decentralized Online Exp-Concave Optimization

Decentralized online convex optimization (D-OCO) is a popular framework for distributed applications with streaming data. To tackle the communication bottleneck, previous studies have investigated D-OCO with compressed communication and proposed several algorithms that are variants of online gradient descent (OGD). However, for D-OCO with exact communication, the best existing algorithms are variants of follow-the-regularized-leader (FTRL). In this paper, for the first time, we propose two FTRL-type algorithms for D-OCO with compressed communication. Compared with OGD-type algorithms, our algorithms are more elegant in both algorithmic design and theoretical analysis. The key insight is that the dual update mechanism of FTRL allows us to make a simple application of the technique for average consensus with communication compression. More specifically, our first algorithm considers the full-information setting, and can match the existing regret bounds. Our second algorithm is designed for the bandit setting, and can significantly improve both the regret bounds and communication costs of existing algorithms.

Distributed online convex optimization (D-OCO) is a powerful paradigm for modeling distributed scenarios with streaming data. However, the communication cost between local learners and the central server is substantial in large-scale applications. To alleviate this bottleneck, we initiate the study of D-OCO with compressed communication. Firstly, to quantify the compression impact, we establish the $Ω(δ^{-1/2}\sqrt{T})$ and $Ω(δ^{-1}\log{T})$ lower bounds for convex and strongly convex loss functions, respectively, where $δ\in (0,1]$ is the compression ratio. Secondly, we propose an optimal algorithm, which enjoys regret bounds of $O(δ^{-1/2}\sqrt{T})$ and $O(δ^{-1} \log T)$ for convex and strongly convex loss functions, respectively. Our method incorporates the error feedback mechanism into the Follow-the-Regularized-Leader framework to address the coupling between the compression error and the projection error. Furthermore, we employ the online compression strategy to mitigate the accumulated error arising from the bidirectional compression. Our online method has great generality, and can be extended to the offline stochastic setting via online-to-batch conversion. We establish convergence rates of $O(δ^{-1/2}T^{-1/2})$ and $O(δ^{-1} T^{-1})$ for convex and strongly convex loss functions, respectively, providing the first guarantees for distributed non-smooth optimization with compressed communication and domain constraints.

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.