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Distributed online constrained convex optimization with event-triggered communication

Kunpeng Zhang, Xinlei Yi, Yuzhe Li, Ming Cao, Tianyou Chai, Tao Yang

TL;DR

The paper addresses distributed online convex optimization with time-varying inequality constraints over a directed, time-varying network. It proposes a distributed event-triggered online primal--dual algorithm that reduces communication by broadcasting decisions only when the change exceeds a nonincreasing threshold, while maintaining sublinear dynamic regret and network constraint violation under appropriate step-size and threshold design. The key contributions include dynamic regret and CCV bounds that adapt to the event-trigger threshold, two corollaries for common threshold decays, and a refinement (Theorem 2) that decouples the threshold from primal updates to achieve sharp sublinear rates. A numerical example confirms the theoretical trade-offs between communication savings (fewer triggers) and performance metrics, highlighting practical relevance for dynamic, constrained multi-agent systems.

Abstract

This paper focuses on the distributed online convex optimization problem with time-varying inequality constraints over a network of agents, where each agent collaborates with its neighboring agents to minimize the cumulative network-wide loss over time. To reduce communication overhead between the agents, we propose a distributed event-triggered online primal-dual algorithm over a time-varying directed graph. With several classes of appropriately chose decreasing parameter sequences and non-increasing event-triggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. Finally, a numerical simulation example is provided to verify the theoretical results.

Distributed online constrained convex optimization with event-triggered communication

TL;DR

The paper addresses distributed online convex optimization with time-varying inequality constraints over a directed, time-varying network. It proposes a distributed event-triggered online primal--dual algorithm that reduces communication by broadcasting decisions only when the change exceeds a nonincreasing threshold, while maintaining sublinear dynamic regret and network constraint violation under appropriate step-size and threshold design. The key contributions include dynamic regret and CCV bounds that adapt to the event-trigger threshold, two corollaries for common threshold decays, and a refinement (Theorem 2) that decouples the threshold from primal updates to achieve sharp sublinear rates. A numerical example confirms the theoretical trade-offs between communication savings (fewer triggers) and performance metrics, highlighting practical relevance for dynamic, constrained multi-agent systems.

Abstract

This paper focuses on the distributed online convex optimization problem with time-varying inequality constraints over a network of agents, where each agent collaborates with its neighboring agents to minimize the cumulative network-wide loss over time. To reduce communication overhead between the agents, we propose a distributed event-triggered online primal-dual algorithm over a time-varying directed graph. With several classes of appropriately chose decreasing parameter sequences and non-increasing event-triggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. Finally, a numerical simulation example is provided to verify the theoretical results.
Paper Structure (11 sections, 8 theorems, 74 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 8 theorems, 74 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation and motivation
  3. Distributed event-triggered online primal--dual algorithm
  4. Algorithm description
  5. Performance metrics
  6. Performance analysis
  7. Numerical example
  8. Conclusions
  9. Preliminary Lemmas
  10. Proof of Theorem 1
  11. Proof of Theorem 2

Key Result

Theorem 1

Suppose Assumptions 1--3 hold. Let $\{ {{x_{i,t}}} \}$ be the sequences generated by Algorithm 1 with where ${\Psi _t} = \sum\nolimits_{{\color{blue} s} = 1}^t {{\tau _{\color{blue} s}}}$, $\kappa \in ( {0,1} )$ are constants. Then, for any $T \in {\mathbb{N}_ + }$ and any comparator sequence ${y_{[ T ]}} \in {\mathcal{X}_T}$, where ${P_T} = \sum\nolimits_{t = 1}^{T - 1} {\| {{y_{t + 1}} - {y_t

Figures (3)

  • Figure 1: Evolutions of $\frac{1}{n}\sum\nolimits_{i = 1}^n {\sum\nolimits_{t = 1}^T {{f_t}( {{x_{i,t}}} )} } /T$.
  • Figure 2: Evolutions of $\frac{1}{n}\sum\nolimits_{i = 1}^n {\sum\nolimits_{t = 1}^T {\| {{{[ {{g_t}( {{x_{i,t}}} )} ]}_ + }} \|} } /T$.
  • Figure 3: Evolutions of total number of triggers.

Theorems & Definitions (22)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Corollary 1
  • Remark 5
  • Corollary 2
  • Remark 6
  • Theorem 2
  • ...and 12 more