[Paper Review] A Resilient Convex Combination for consensus-based distributed algorithms

Xuan Wang, Shaoshuai Mou+1 more|arXiv (Cornell University)|Jun 27, 2018

Distributed Control Multi-Agent Systems23 references3 citations

TL;DR

This paper proposes a low-computational-complexity method for resilient convex combination in consensus-based distributed algorithms by leveraging the intersection of convex hulls to isolate normal vectors from malicious ones. The approach enables consensus under Byzantine attacks without requiring prior identification of malicious agents, outperforming Tverberg point-based methods in convergence speed and applicability to time-varying networks.

ABSTRACT

Consider a set of vectors in $\mathbb{R}^n$, partitioned into two classes: normal vectors and malicious vectors. The number of malicious vectors is bounded but their identities are unknown. The paper provides a way for achieving a resilient convex combination, which is a convex combination of only normal vectors. Compared with existing approaches based on Tverberg points, the proposed method based on the intersection of convex hulls has lower computational complexity. Simulations suggest that the proposed method can be applied to resilience for consensus-based distributed algorithms against Byzantine attacks.

Motivation & Objective

• To address the challenge of achieving resilient consensus in multi-agent systems under Byzantine attacks where malicious agents inject false information.
• To develop a method for convex combination that excludes malicious vectors without prior identification of their identities.
• To reduce computational complexity compared to existing Tverberg point-based approaches for resilient aggregation.
• To enable resilience in both fixed and time-varying network topologies, including constrained consensus problems.
• To validate the method’s effectiveness through simulations in unconstrained and constrained consensus scenarios.

Proposed method

• The method formulates the resilient convex combination as a vector within the intersection of multiple convex hulls formed from subsets of the agent states.
• It models the set of valid resilient combinations as a set of linear constraints, enabling solution via quadratic programming.
• The algorithm computes a convex combination that lies within the intersection of convex hulls of all possible subsets of size at least p, where p is the lower bound on normal vectors.
• The approach avoids the need to identify malicious agents by relying solely on the geometric property that the intersection of convex hulls of sufficiently large subsets contains only normal vectors.
• The resilient combination is used to replace standard convex combinations in consensus update rules, ensuring convergence despite malicious inputs.
• The method is applicable to both unconstrained and constrained consensus problems, such as solving linear equations in distributed settings.

Experimental results

Research questions

• RQ1 Can a resilient convex combination be computed with lower computational complexity than Tverberg point-based methods?
• RQ2 Can the proposed method ensure consensus in the presence of Byzantine agents without identifying them?
• RQ3 Does the method remain effective under time-varying network topologies where Tverberg-based methods fail?
• RQ4 How does the convergence speed of the proposed method compare to Tverberg point-based resilient aggregation?
• RQ5 Can the resilient convex combination be effectively integrated into distributed algorithms for solving linear equations under adversarial conditions?

Key findings

• The proposed method achieves resilient convex combination through intersection of convex hulls, reducing computational complexity compared to Tverberg point-based approaches.
• Simulations show that the resilient convex combination enables consensus in both unconstrained and constrained consensus problems even with malicious agents injecting false data.
• The method outperforms Tverberg point-based resilient aggregation in convergence speed, as demonstrated in simulations with fixed and time-varying networks.
• The approach is effective in time-varying networks where Tverberg-based methods fail due to insufficient neighbor counts.
• The resilient convex combination ensures exponential convergence to the correct consensus value in distributed linear equation solving, even under Byzantine attacks.
• The method does not require identification of malicious agents, making it suitable for dynamic and adversarial environments.