[Paper Review] A "Safe Kernel" Approach for Resilient Multi-Dimensional Consensus

Jiaqi Yan, Yilin Mo+2 more|arXiv (Cornell University)|Nov 25, 2019

Distributed Control Multi-Agent Systems19 references4 citations

TL;DR

This paper proposes a resilient multi-dimensional consensus algorithm using a 'safe kernel' technique, where healthy agents iteratively adjust their states toward a point within a robustly defined kernel formed from neighbor data. Under sufficient network robustness conditions, benign agents exponentially converge to a consensus within the convex hull of their initial states, even under adversarial attacks, improving accuracy over scalar-based methods.

ABSTRACT

This paper considers the resilient multi-dimensional consensus problem in networked systems, where some of the agents might be malicious (or faulty). We propose a multi-dimensional consensus algorithm, where at each time step each healthy agent computes a "safe kernel" based on the information from its neighbors, and modifies its own state towards a point inside the kernel. Assuming that the number of malicious agents is locally (or globally) upper bounded, sufficient conditions on the network topology are presented to guarantee that the benign agents exponentially reach an agreement within the convex hull of their initial states, regardless of the actions of the misbehaving ones. It is also revealed that the graph connectivity and robustness required to achieve the resilient consensus increases linearly with respect to the dimension of the agents' state, indicating the existence of a trade-off between the low communication cost and system security. Numerical examples are provided in the end to validate the theoretical results.

Motivation & Objective

• To address the resilient multi-dimensional consensus problem in networks with malicious or faulty agents.
• To improve upon scalar-based consensus methods that converge to a box-shaped region, which is less accurate in multi-dimensional settings.
• To design a distributed algorithm that guarantees convergence within the convex hull of benign agents' initial states, enhancing security and accuracy.
• To establish topological conditions—specifically robustness metrics—that ensure resilience against both local and global adversarial attacks.
• To extend the applicability of safe kernel techniques beyond consensus to other distributed problems like optimization and estimation.

Proposed method

• Each healthy agent computes a 'safe kernel' at each time step using its neighbors' states, defined as the intersection of all convex hulls formed by its neighbors and itself.
• The safe kernel is constructed by removing extreme values in each dimension, similar to MSR, but generalized to multi-dimensional vectors.
• Agents update their state by moving toward a point inside the safe kernel, using a convex combination of the kernel's vertices.
• The algorithm ensures exponential convergence to a common value within the convex hull of benign agents' initial states.
• Network robustness is formalized as a graph-theoretic condition: ((d+1)F+1)-robustness for local attacks and (dF+1,F+1)-robustness for global attacks.
• The method is proven to be resilient even under worst-case adversarial behavior, including full knowledge of topology and adaptive attacks.

Experimental results

Research questions

• RQ1 Can a distributed consensus algorithm be designed to ensure convergence within the convex hull of benign agents' initial states in multi-dimensional systems under adversarial conditions?
• RQ2 What network topology conditions are necessary and sufficient to guarantee resilient consensus in multi-dimensional settings with faulty agents?
• RQ3 How does the required robustness of the network scale with the dimensionality of the agents' state vectors?
• RQ4 Can the safe kernel technique outperform naive scalarization of multi-dimensional consensus by applying scalar resilient algorithms to each dimension?
• RQ5 Is it possible to achieve a suboptimal but secure consensus that avoids arbitrary convergence points dictated by malicious agents?

Key findings

• The proposed algorithm guarantees exponential convergence of all benign agents to a common value within the convex hull of their initial states, regardless of malicious actions.
• For F-local attacks, the network must be ((d+1)F+1)-robust; for F-total attacks, it must be (dF+1,F+1)-robust, showing a linear increase in required robustness with dimension d.
• The safe kernel approach ensures convergence to a convex combination of benign initial states, improving accuracy over scalar-based methods that converge to a box-shaped region.
• The algorithm remains effective even when misbehaving agents have full knowledge of the network and update rules, and can send different values to different neighbors.
• Numerical simulations validate the theoretical results, showing that benign agents remain within the convex hull and converge to a common value despite a malicious agent injecting arbitrary signals.
• The work establishes a foundation for extending the safe kernel technique to other distributed problems such as optimization and estimation in adversarial environments.