Distributed Online Convex Optimization with Compressed Communication: Optimal Regret and Applications

2026-04-10

AI Summary

This work addresses the high communication overhead in distributed online convex optimization under large-scale streaming data by studying optimization methods with compressed communication. By integrating the Follow-the-Regularized-Leader framework, an error-feedback mechanism, and a bidirectional online compression strategy—augmented with an online-to-batch conversion—the proposed approach effectively decouples compression errors from projection errors and controls their accumulation. The paper establishes, for the first time, theoretical lower bounds on regret for both convex and strongly convex loss functions under compressed communication, and introduces an optimal algorithm that achieves these bounds. It also provides the first convergence guarantees for distributed nonsmooth optimization with compressed communication and domain constraints. The method attains optimal regret bounds of $O(\delta^{-1/2}\sqrt{T})$ and $O(\delta^{-1}\log T)$ in the convex and strongly convex settings, respectively, yielding corresponding offline convergence rates of $O(\delta^{-1/2}T^{-1/2})$ and $O(\delta^{-1}T^{-1})$.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 $\Omega(\delta^{-1/2}\sqrt{T})$ and $\Omega(\delta^{-1}\log{T})$ lower bounds for convex and strongly convex loss functions, respectively, where $\delta \in (0,1]$ is the compression ratio. Secondly, we propose an optimal algorithm, which enjoys regret bounds of $O(\delta^{-1/2}\sqrt{T})$ and $O(\delta^{-1} \log T)$ 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 $O(\delta^{-1/2}T^{-1/2})$ and $O(\delta^{-1} T^{-1})$ for convex and strongly convex loss functions, respectively, providing the first guarantees for distributed non-smooth optimization with compressed communication and domain constraints.

Problem

• Distributed Online Convex Optimization
• Compressed Communication
• Communication Cost
• Regret Bounds
• Error Feedback

Innovation

• compressed communication
• distributed online convex optimization
• error feedback
• regret bound
• online-to-batch conversion