Distributed Online Convex Optimization with Nonseparable Costs and Constraints

2026-02-11

AI Summary

This work addresses the challenge of jointly optimizing non-separable global cost functions under coupled long-term constraints in distributed online convex optimization. The authors propose a distributed online primal-dual belief consensus algorithm, wherein local agents exchange beliefs about the global decision over a communication graph to collaboratively optimize the non-separable objective and satisfy non-separable long-term constraints under full-information feedback. The key innovation lies in a belief-sharing mechanism that decouples the consensus error of primal variables from dual constraint violations, enabling—for the first time in the non-separable setting—a sublinear bound on cumulative constraint violations. Theoretical analysis shows that the algorithm simultaneously achieves $O(T^{1/2})$ regret and cumulative constraint violation bounds over a time horizon $T$, breaking the existing $O(T^{3/4})$ barrier and matching the known lower bound for online constrained convex optimization.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper studies distributed online convex optimization with time-varying coupled constraints, motivated by distributed online control in network systems. Most prior work assumes a separability condition: the global objective and coupled constraint functions are sums of local costs and individual constraints. In contrast, we study a group of agents, networked via a communication graph, that collectively select actions to minimize a sequence of nonseparable global cost functions and to stratify nonseparable long-term constraints based on full-information feedback and intra-agent communication. We propose a distributed online primal-dual belief consensus algorithm, where each agent maintains and updates a local belief of the global collective decisions, which are repeatedly exchanged with neighboring agents. Unlike the previous consensus primal-dual algorithms under separability that ask agents to only communicate their local decisions, our belief-sharing protocol eliminates coupling between the primal consensus disagreement and the dual constraint violation, yielding sublinear regret and cumulative constraint violation (CCV) bounds, both in $O({T}^{1/2})$, where $T$ denotes the time horizon. Such a result breaks the long-standing $O(T^{3/4})$ barrier for CCV and matches the lower bound of online constrained convex optimization, indicating the online learning efficiency at the cost of communication overhead.

Problem

• distributed online optimization
• nonseparable costs
• coupled constraints
• online convex optimization
• networked agents

Innovation

• distributed online optimization
• nonseparable constraints
• primal-dual algorithm
• belief consensus
• sublinear regret