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Fast Distributed Optimization over Directed Graphs under Malicious Attacks using Trust

Arif Kerem Dayı, Orhan Eren Akgün, Stephanie Gil, Michal Yemini, Angelia Nedić

TL;DR

This paper tackles fast distributed optimization on directed graphs in the presence of malicious agents. It presents RP3, a resilient variant of the Projected Push-Pull method that uses inter-agent trust and gradient tracking, with growing constraint sets to bound adversarial influence until trust estimates stabilize. Theoretical results establish almost-sure and $r$-th mean convergence to the nominal optimum, plus geometric convergence in expectation under appropriate step sizes and set growth; the unbounded-set extension broadens applicability to unconstrained problems. Numerical experiments on constrained consensus and multi-robot target tracking validate RP3's resilience and faster convergence relative to trust-based and data-filtering baselines, highlighting its practical impact for cyber-physical systems under attack.

Abstract

In this work, we introduce the Resilient Projected Push-Pull (RP3) algorithm designed for distributed optimization in multi-agent cyber-physical systems with directed communication graphs and the presence of malicious agents. Our algorithm leverages stochastic inter-agent trust values and gradient tracking to achieve geometric convergence rates in expectation even in adversarial environments. We introduce growing constraint sets to limit the impact of the malicious agents without compromising the geometric convergence rate of the algorithm. We prove that RP3 converges to the nominal optimal solution almost surely and in the $r$-th mean for any $r\geq 1$, provided the step sizes are sufficiently small and the constraint sets are appropriately chosen. We validate our approach with numerical studies on average consensus and multi-robot target tracking problems, demonstrating that RP3 effectively mitigates the impact of malicious agents and achieves the desired geometric convergence.

Fast Distributed Optimization over Directed Graphs under Malicious Attacks using Trust

TL;DR

This paper tackles fast distributed optimization on directed graphs in the presence of malicious agents. It presents RP3, a resilient variant of the Projected Push-Pull method that uses inter-agent trust and gradient tracking, with growing constraint sets to bound adversarial influence until trust estimates stabilize. Theoretical results establish almost-sure and -th mean convergence to the nominal optimum, plus geometric convergence in expectation under appropriate step sizes and set growth; the unbounded-set extension broadens applicability to unconstrained problems. Numerical experiments on constrained consensus and multi-robot target tracking validate RP3's resilience and faster convergence relative to trust-based and data-filtering baselines, highlighting its practical impact for cyber-physical systems under attack.

Abstract

In this work, we introduce the Resilient Projected Push-Pull (RP3) algorithm designed for distributed optimization in multi-agent cyber-physical systems with directed communication graphs and the presence of malicious agents. Our algorithm leverages stochastic inter-agent trust values and gradient tracking to achieve geometric convergence rates in expectation even in adversarial environments. We introduce growing constraint sets to limit the impact of the malicious agents without compromising the geometric convergence rate of the algorithm. We prove that RP3 converges to the nominal optimal solution almost surely and in the -th mean for any , provided the step sizes are sufficiently small and the constraint sets are appropriately chosen. We validate our approach with numerical studies on average consensus and multi-robot target tracking problems, demonstrating that RP3 effectively mitigates the impact of malicious agents and achieves the desired geometric convergence.
Paper Structure (30 sections, 21 theorems, 74 equations, 2 figures, 1 algorithm)

This paper contains 30 sections, 21 theorems, 74 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Formulation
  4. Notation
  5. Problem Setup
  6. Trust Opinions
  7. Problem Definition
  8. Resilient Projected Push-Pull (RP3) Algorithm
  9. Background
  10. Algorithm
  11. Trust-based weights
  12. Preserving the gradient tracking property in the presence of malicious agents
  13. Projecting the gradient tracking variables onto the set $\mathcal{S}_k$
  14. Analysis
  15. Preliminary Results: Learning the Trustworthiness of the Agents
  16. ...and 15 more sections

Key Result

Theorem 1

Define the error vector $\mathbf{e}[k]=(\left\|\mathbf{x}[k] - \mathbf{x}^*\right\|_{\phi}, D(\mathbf{x}[k], \phi), S(\mathbf{y}[k], \pi))^\intercal.$ Let assumption:cost_function, assumption:closed_convex_const_set, and assumption:graph-connectivity hold. Let $0<\eta<\frac{1}{nL}$ and where Then, we have where the inequality is elementwise and $M(\eta, \lambda) \in \mathbb{R}^{3\times 3}$ is e

Figures (2)

  • Figure 1: Constrained consensus experiment results with $L=50$ legitimate and $M=100$ malicious agents.
  • Figure 2: Target tracking experiment results with $L=9$ legitimate and $M=6$ malicious agents. (a) Convergence to the optimal trajectory. (b) Visualization of the final trajectories for all methods.

Theorems & Definitions (47)

  • Definition 1: Growth of the set sequence $\{\mathcal{X}_k \}$
  • Definition 2: Projection onto $\mathcal{X}$
  • Definition 3: Stochastic Observation of Trust $\alpha_{ij}$
  • Definition 4: Opinion of Trust
  • Theorem 1: Theorem 1, projected_push_pull
  • Definition 5: The nominal behavior of the RP3
  • Lemma 1: L4DC, Corollary 1
  • Corollary 1
  • Proposition 1: Proposition 1, L4DC_2023_extended
  • Corollary 2: Boundedness of Gradients
  • ...and 37 more