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Distributed fixed-point algorithms for dynamic convex optimization over decentralized and unbalanced wireless networks

Navneet Agrawal, Renato L. G. Cavalcante, Sławomir Stańczak

TL;DR

The paper addresses distributed fixed-point optimization over decentralized, time-varying directed wireless networks by introducing a unified ATC framework that couples local processing with consensus via quasi-Fejér monotone operators. It develops two key strands: (i) a general convergence theory for directed-graph communication, establishing almost-sure convergence to a common estimate and, under favorable interiority conditions, consensus to a single point; (ii) a dynamic-optimization approach using sAPSM that handles time-varying objectives and yields convergence to a time-invariant solution of an evolving sequence of convex problems. A novel OTA-C consensus protocol is proposed to reduce communication overhead and increase network autonomy while preserving the convergence guarantees, and the framework is demonstrated on a distributed supervised-learning task over time-varying wireless networks with notable latency and energy-efficiency gains. Overall, the work provides a versatile toolkit for designing and analyzing distributed solvers that leverage advanced communication and optimization techniques in realistic, time-varying network settings.

Abstract

We consider problems where agents in a network seek a common quantity, measured independently and periodically by each agent through a local time-varying process. Numerous solvers addressing such problems have been developed in the past, featuring various adaptations of the local processing and the consensus step. However, existing solvers still lack support for advanced techniques, such as superiorization and over-the-air function computation (OTA-C). To address this limitation, we introduce a comprehensive framework for the analysis of distributed algorithms by characterizing them using the quasi-Fejér type algorithms and an extensive communication model. Under weak assumptions, we prove almost sure convergence of the algorithm to a common estimate for all agents. Moreover, we develop a specific class of algorithms within this framework to tackle distributed optimization problems with time-varying objectives, and, assuming that a time-invariant solution exists, prove its convergence to a solution. We also present a novel OTA-C protocol for consensus step in large decentralized networks, reducing communication overhead and enhancing network autonomy as compared to the existing protocols. The effectiveness of the algorithm, featuring superiorization and OTA-C, is demonstrated in a real-world application of distributed supervised learning over time-varying wireless networks, highlighting its low-latency and energy-efficiency compared to standard approaches.

Distributed fixed-point algorithms for dynamic convex optimization over decentralized and unbalanced wireless networks

TL;DR

The paper addresses distributed fixed-point optimization over decentralized, time-varying directed wireless networks by introducing a unified ATC framework that couples local processing with consensus via quasi-Fejér monotone operators. It develops two key strands: (i) a general convergence theory for directed-graph communication, establishing almost-sure convergence to a common estimate and, under favorable interiority conditions, consensus to a single point; (ii) a dynamic-optimization approach using sAPSM that handles time-varying objectives and yields convergence to a time-invariant solution of an evolving sequence of convex problems. A novel OTA-C consensus protocol is proposed to reduce communication overhead and increase network autonomy while preserving the convergence guarantees, and the framework is demonstrated on a distributed supervised-learning task over time-varying wireless networks with notable latency and energy-efficiency gains. Overall, the work provides a versatile toolkit for designing and analyzing distributed solvers that leverage advanced communication and optimization techniques in realistic, time-varying network settings.

Abstract

We consider problems where agents in a network seek a common quantity, measured independently and periodically by each agent through a local time-varying process. Numerous solvers addressing such problems have been developed in the past, featuring various adaptations of the local processing and the consensus step. However, existing solvers still lack support for advanced techniques, such as superiorization and over-the-air function computation (OTA-C). To address this limitation, we introduce a comprehensive framework for the analysis of distributed algorithms by characterizing them using the quasi-Fejér type algorithms and an extensive communication model. Under weak assumptions, we prove almost sure convergence of the algorithm to a common estimate for all agents. Moreover, we develop a specific class of algorithms within this framework to tackle distributed optimization problems with time-varying objectives, and, assuming that a time-invariant solution exists, prove its convergence to a solution. We also present a novel OTA-C protocol for consensus step in large decentralized networks, reducing communication overhead and enhancing network autonomy as compared to the existing protocols. The effectiveness of the algorithm, featuring superiorization and OTA-C, is demonstrated in a real-world application of distributed supervised learning over time-varying wireless networks, highlighting its low-latency and energy-efficiency compared to standard approaches.
Paper Structure (11 sections, 7 theorems, 50 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General class of distributed algorithms
  3. System description and agent dynamics
  4. Quasi-Fejérian characterization of ${(\mathsf{T}_{k, i})_{i\in\mathbb{N}}}$ and analysis
  5. Dynamic distributed convex optimization via sAPSM
  6. OTA-C for decentralized consensus
  7. Distributed machine learning application
  8. Conclusion
  9. Proof of Theorem 1
  10. Proof of Theorem 2
  11. Proof of Proposition 1

Key Result

Theorem 1

Suppose that Assumption as:net holds in a system where each agent ${k\in\mathcal{A}}$ implements the scheme in eq:agent_scheme, where $\mathsf{K}_{k,i}$ is given by eq:comm_model, and, for all ${k\in\mathcal{A}}$, $(\mathsf{T}_{k, i})_{i\in\mathbb{N}} \equiv (\mathsf{T}_{k, i}^{(\mathcal{Q}_k)})_{i\ (iii) (Characterization of accumulation points): In addition, assume that the set $\mathcal{Q}^\sta

Figures (1)

  • Figure 1: Estimation error in terms of NMSE

Theorems & Definitions (19)

  • Remark 1
  • Definition 1: $(\mathsf{T}_{i}^{(\mathcal{Q})})_{i\in\mathbb{N}}$ : Quasi-Fejér monotone sequence (QFMS) generator
  • Theorem 1
  • proof
  • Remark 2
  • Definition 2: $(\mathsf{T}_{i}^{(\Theta)})_{i\in\mathbb{N}}$ : Superiorized APSM (sAPSM) sequence generator
  • Theorem 2
  • proof
  • Remark 3
  • Proposition 1
  • ...and 9 more