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Distributed Online Convex Optimization with Nonseparable Costs and Constraints

Zhaoye Pan, Haozhe Lei, Fan Zuo, Zilin Bian, Tao Li

TL;DR

This work addresses distributed online convex optimization with time-varying, globally coupled (nonseparable) costs and constraints. It introduces a belief-consensus-based distributed online primal-dual algorithm (dopbc) in which agents maintain and exchange beliefs about joint actions and multipliers, enabling decentralized gradient-like updates. Theoretical results show sublinear performance: static regret and cumulative constraint violation both scale as $O(\\sqrt{T})$ under a fixed step size by selecting $c=1/2$ in $\alpha_t=T^{-c}$, breaking the prior $O(T^{3/4})$ CCV barrier. The key insight is that belief sharing decouples primal consensus disagreement from dual constraint violation, achieving optimal online learning efficiency at the cost of additional communication.

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.

Distributed Online Convex Optimization with Nonseparable Costs and Constraints

TL;DR

This work addresses distributed online convex optimization with time-varying, globally coupled (nonseparable) costs and constraints. It introduces a belief-consensus-based distributed online primal-dual algorithm (dopbc) in which agents maintain and exchange beliefs about joint actions and multipliers, enabling decentralized gradient-like updates. Theoretical results show sublinear performance: static regret and cumulative constraint violation both scale as under a fixed step size by selecting in , breaking the prior CCV barrier. The key insight is that belief sharing decouples primal consensus disagreement from dual constraint violation, achieving optimal online learning efficiency at the cost of additional communication.

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 , where denotes the time horizon. Such a result breaks the long-standing barrier for CCV and matches the lower bound of online constrained convex optimization, indicating the online learning efficiency at the cost of communication overhead.
Paper Structure (13 sections, 5 theorems, 37 equations, 1 table, 1 algorithm)

This paper contains 13 sections, 5 theorems, 37 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Works
  4. Problem Formulation
  5. Distributed Online Primal-Dual with Belief Consensus
  6. Assumptions
  7. Theoretic Analysis
  8. Conclusion
  9. Proof of Lemma \ref{['lemma:consensus_recur']}
  10. Proof of Lemma \ref{['lemma:drift']}
  11. Proof of Theorem \ref{['thm:violation']}
  12. Proof of Theorem \ref{['thm:static_regret']}
  13. Derivation of Inequality \ref{['eq:coupling_bound']}

Key Result

Lemma 1

Under Assumptions ass:std and ass:net, for any constant $\beta>0$, the primal consensus error $\delta_{x,t}^2$ generated by Algorithm alg:dopbc satisfies $\delta_{x,t+1}^2 \le (1+\beta)\sigma^2 \delta_{x,t}^2 + (1+1/\beta)\alpha^2 N G_d^2,$ where $G_d \triangleq G_f + \Lambda_{\max} G_g$.

Theorems & Definitions (9)

  • Definition 1: Static Regret
  • Definition 2: Cumulative Constraint Violation
  • Remark 1: Justification for Bounded Dual Domain
  • Lemma 1: One-step Primal Consensus Error
  • Lemma 2: One-Step Primal-Dual Drift Analysis
  • Theorem 1: Cumulative Constraint Violation
  • Theorem 2: Static Regret
  • Proposition 1: Primal-Dual Decoupling
  • Remark 2