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A Dynamically Weighted ADMM Framework for Byzantine Resilience

Vishnu Vijay, Kartik A. Pant, Minhyun Cho, Inseok Hwang

TL;DR

This work addresses resilience of distributed optimization under Byzantine faults in ADMM. It introduces Dynamically Weighted ADMM (DW-ADMM), which uses time-varying edge weights and Laplacian updates to dampen adversarial influence while preserving consensus. The authors prove error-free convergence to the global optimum and, under Byzantine threats, bounded convergence to stationary points, supported by rigorous lemmas and simulations. Numerical experiments on a 10-node network demonstrate DW-ADMM matching conventional ADMM when clean and maintaining bounded behavior under malicious perturbations, highlighting practical resilience benefits.

Abstract

The alternating direction of multipliers method (ADMM) is a popular method to solve distributed consensus optimization utilizing efficient communication among various nodes in the network. However, in the presence of faulty or attacked nodes, even a small perturbation (or sharing false data) during the communication can lead to divergence of the solution. To address this issue, in this work we consider ADMM under the effect of Byzantine threat, where an unknown subset of nodes is subject to Byzantine attacks or faults. We propose Dynamically Weighted ADMM (DW-ADMM), a novel variant of ADMM that uses dynamic weights on the edges of the network, thus promoting resilient distributed optimization. We establish that the proposed method (i) produces a nearly identical solution to conventional ADMM in the error-free case, and (ii) guarantees a bounded solution with respect to the global minimizer, even under Byzantine threat. Finally, we demonstrate the effectiveness of our proposed algorithm using an illustrative numerical simulation.

A Dynamically Weighted ADMM Framework for Byzantine Resilience

TL;DR

This work addresses resilience of distributed optimization under Byzantine faults in ADMM. It introduces Dynamically Weighted ADMM (DW-ADMM), which uses time-varying edge weights and Laplacian updates to dampen adversarial influence while preserving consensus. The authors prove error-free convergence to the global optimum and, under Byzantine threats, bounded convergence to stationary points, supported by rigorous lemmas and simulations. Numerical experiments on a 10-node network demonstrate DW-ADMM matching conventional ADMM when clean and maintaining bounded behavior under malicious perturbations, highlighting practical resilience benefits.

Abstract

The alternating direction of multipliers method (ADMM) is a popular method to solve distributed consensus optimization utilizing efficient communication among various nodes in the network. However, in the presence of faulty or attacked nodes, even a small perturbation (or sharing false data) during the communication can lead to divergence of the solution. To address this issue, in this work we consider ADMM under the effect of Byzantine threat, where an unknown subset of nodes is subject to Byzantine attacks or faults. We propose Dynamically Weighted ADMM (DW-ADMM), a novel variant of ADMM that uses dynamic weights on the edges of the network, thus promoting resilient distributed optimization. We establish that the proposed method (i) produces a nearly identical solution to conventional ADMM in the error-free case, and (ii) guarantees a bounded solution with respect to the global minimizer, even under Byzantine threat. Finally, we demonstrate the effectiveness of our proposed algorithm using an illustrative numerical simulation.

Paper Structure

This paper contains 12 sections, 6 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations
  4. Network Model
  5. Byzantine Threat Model
  6. Distributed Optimization
  7. Proposed Algorithm Development
  8. Main Results
  9. Problem Assumptions and Supporting Lemmas
  10. Convergence and Boundedness
  11. Simulation Results
  12. Conclusion

Key Result

Lemma 1

The following are equivalent to the dual variable update step in eq:prop_lam_update, where $N = ( L_-^0 )^{1/2}$ and $Y^0 = 0$:

Figures (2)

  • Figure 1: Network Topology for Simulations
  • Figure 2: Left Convergence without Error, Center Boundedness with Ramp Error, Right Boundedness with Random Error.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1: Error-Free Convergence
  • proof
  • ...and 2 more