A Distributed Algorithm for Solving a Linear Algebraic Equation (1503.00808v1)

Abstract: A distributed algorithm is described for solving a linear algebraic equation of the form $Ax=b$ assuming the equation has at least one solution. The equation is simultaneously solved by $m$ agents assuming each agent knows only a subset of the rows of the partitioned matrix $(A,b)$, the current estimates of the equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the equation has a solution and any sequence of "repeatedly jointly strongly connected graphs" $\mathbb{N}(t)$, $t=1,2,\ldots$, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax=b$. It is also shown that the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax=b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$, even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax=b$ in a distributed manner, even if $Ax=b$ does not have a solution.

• The paper introduces a distributed algorithm that ensures exponential convergence of solutions when agents use jointly strongly connected, time-dependent communication graphs.
• The paper demonstrates that each agent, with partial data, collaborates and updates asynchronously to rapidly eliminate discrepancies in their local solutions.
• The paper reveals the algorithm's adaptability to real-time changes in system parameters, including extending to least squares solutions when a direct solution of Ax = b is not possible.

Distributed Algorithm for Solving Linear Algebraic Equations

The paper presents a distributed algorithm for solving linear algebraic equations of the form $Ax = b$. The central focus of this research is on developing a method that enables multiple agents, each possessing only partial information about the matrix $A$ and vector $b$, to collaboratively and efficiently find a solution $\mathbf{x}$ that satisfies the equation. Here, each agent knows a subset of rows of $A$, the corresponding subset of vector $b$, and receives solution estimates from its neighboring agents. The communicative relations among agents are structured as a time-dependent directed graph, $\mathbb{N}(t)$.

The core contribution of this work is a proof that the proposed algorithm leads to exponential convergence of the estimated solutions from all agents to the true solution under the condition that the directed graph sequences are "repeatedly jointly strongly connected." The necessity of this condition is emphasized for achieving exponential convergence, which is paramount for practical applicability, especially in dynamic and asynchronous network scenarios.

Key Findings

1. Algorithm Performance:
   • The convergence to the correct solution occurs exponentially fast if the communication graph among the agents, as represented over time by the sequence of directed graphs $\mathbb{N}(t)$, is repeatedly jointly strongly connected.
   • This condition ensures that, over time, all agents effectively exchange the information necessary to eliminate discrepancies in their local solutions, rapidly converging to a single, consensus solution.
2. Graph Properties and Requirements:
   • The convergence condition's necessity and sufficiency are detailed, with proofs indicating that the absence of this connectivity property precludes the guarantee of exponential convergence.
   • The paper further establishes that if this graph sequence condition is met, each agent’s state, updated asynchronously, will also converge exponentially to the solution, even in scenarios without synchronized timing.
3. Adaptation to Changing Systems:
   • The algorithm is robust to minor changes in $A$ and $b$, capable of adapting the solution in real-time as these changes occur, provided they evolve slowly.
   • This adaptability implies the method’s relevance in dynamic environments where $A$ and $b$ could change, such as in real-time sensor networks or adaptive systems.
4. Algorithmic Extensions:
   • A variant of the algorithm is proposed for the cases where $Ax = b$ does not have a solution, aiming to find a least squares solution instead.

Implications

• Practical Application: Exponential convergence means that this algorithm can be employed in practical distributed systems, notably where real-time decision-making based on collaborative computation is crucial, such as in distributed control systems, sensor fusion in networks, and other cooperative multi-agent systems.
• Theoretical Insight: The work provides significant theoretical insights into the dynamics of distributed consensus algorithms, applicable to a broader class of distributed optimization problems.
• Future Directions: Future research could explore reducing the communication overhead and computational load required per agent, enhancing robustness to more significant changes in $A$ and $b$, and extending classes of problems beyond linear equations.

Overall, this paper makes a substantive contribution to the field of distributed computation, paving the way for further exploration in collaborative algorithm design that leverages the processing power of decentralized agents to solve complex algebraic problems efficiently.