[Paper Review] A Double-Layered Framework for Distributed Coordination in Solving Linear Equations

Xuan Wang, Shaoshuai Mou+1 more|arXiv (Cornell University)|Nov 29, 2017

Distributed Control Multi-Agent Systems28 references3 citations

TL;DR

This paper proposes a double-layered multi-agent framework that integrates consensus and conservation mechanisms to solve large-scale linear equations in a distributed manner. By organizing agents into clusters where intra-cluster communication achieves consensus and inter-cluster communication enforces conservation, the method enables each agent to operate with minimal state (as few as two scalars) and without requiring full knowledge of the global equation matrix, achieving exponential convergence with heterogeneous state dimensions.

ABSTRACT

This paper proposes a double-layered framework (or form of network) to integrate two mechanisms, termed consensus and conservation, achieving distributed solution of a linear equation. The multi-agent framework considered in the paper is composed of clusters (which serve as a form of aggregating agent) and each cluster consists of a sub-network of agents. By achieving consensus and conservation through agent-agent communications in the same cluster and cluster-cluster communications, distributed algorithms are devised for agents to cooperatively achieve a solution to the overall linear equation. These algorithms outperform existing consensus-based algorithms, including but not limited to the following aspects: first, each agent does not have to know as much as a complete row or column of the overall equation; second, each agent only needs to control as few as two scalar states when the number of clusters and the number of agents are sufficiently large; third, the dimensions of agents' states in the proposed algorithms do not have to be the same (while in contrast, algorithms based on the idea of standard consensus inherently require all agents' states to be of the same dimension). Both analytical proof and simulation results are provided to validate exponential convergence of the proposed distributed algorithms in solving linear equations.

Motivation & Objective

• To address the limitations of existing consensus-based distributed solvers that require agents to know full rows/columns of the global matrix.
• To overcome the constraint of uniform state dimensionality in standard consensus algorithms.
• To integrate both consensus (for agreement) and conservation (for sum-invariance) in a hierarchical network structure.
• To enable distributed solution of linear equations with minimal agent state complexity and scalable communication.
• To achieve exponential convergence without requiring all agents to have identical state dimensions.

Proposed method

• The framework uses a two-layer architecture: clusters of agents (intra-cluster communication) and inter-cluster communication between clusters.
• Intra-cluster communication enforces consensus among agents within each cluster to align their states.
• Inter-cluster communication enforces conservation by ensuring the sum of certain functions of agent states remains constant across clusters.
• The method employs a hybrid update law combining local consensus dynamics and global conservation constraints via a Lyapunov-based design.
• Agents maintain scalar or low-dimensional state vectors, with state dimensionality independent across agents.
• The algorithm is proven to converge exponentially by analyzing the eigenvalue properties of the system matrix using a transformed symmetric negative semi-definite form.

Experimental results

Research questions

• RQ1 Can a distributed algorithm solve a large linear system without requiring each agent to know the full row or column of the global matrix?
• RQ2 How can consensus and conservation be jointly enforced in a multi-agent network to solve linear equations?
• RQ3 Can agents operate with heterogeneous state dimensions while still achieving global convergence?
• RQ4 What is the minimal state dimension required per agent for exponential convergence in such a distributed framework?
• RQ5 How does the double-layered architecture improve convergence speed and scalability compared to single-layer consensus-based solvers?

Key findings

• The proposed algorithm achieves exponential convergence to the true solution of the linear system, as proven via eigenvalue analysis of the system matrix.
• Each agent only needs to maintain as few as two scalar states when the number of clusters and agents matches the dimensions of the linear system.
• The method allows agents to have different state dimensions, breaking the fundamental constraint of standard consensus-based algorithms requiring uniform state dimensions.
• Agents do not need to know the full row or column of the global matrix, reducing information requirements and improving privacy and scalability.
• The double-layered structure enables efficient coordination: consensus within clusters and conservation across clusters, leading to faster convergence than existing methods.
• Theoretical analysis confirms that the zero eigenvalue of the system matrix is non-defective, ensuring stable and convergent behavior.