Security of Gradient Tracking Algorithms Against Malicious Agents

Abstract

Consensus algorithms are fundamental to multi-agent distributed optimization, and their security under adversarial conditions is an active area of research. While prior works primarily establish conditions for successful global consensus under attack, little is known about system behavior when these conditions are violated. This paper addresses this gap by investigating the robustness of the Wang--Elia algorithm, which is a robust to noise version of gradient tracking algorithm, in the presence of malicious agents. We consider a network of agents collaboratively minimizing a global cost function, where a subset of agents may transmit faulty information to disrupt consensus. To quantify resilience, we formulate a security metric as an optimization problem, which is rooted in centralized attack detection literature. We provide a tractable reformulation of the optimization problem, and derive conditions under which the metric becomes unbounded, identifying undetectable attack signals that reveal inherent vulnerabilities. To facilitate design and analysis, we propose a well-posed variant of the metric and propose design methods to enhance network robustness against stealthy adversarial attacks. Numerical examples demonstrate the effectiveness of the proposed framework to enhance the resilience of multi-agent distributed optimization.