[论文解读] A Distributed Algorithm for Solving a Linear Algebraic Equation

Shaoshuai Mou, Ji Liu+1 more|arXiv (Cornell University)|Mar 3, 2015

Distributed Control Multi-Agent Systems参考文献 35被引用 8

一句话总结

该论文提出了一种分布式算法，用于在由 $m$ 个代理组成的网络中求解线性系统 $Ax = b$，其中每个代理仅持有 $[A\;b]$ 的部分行，并仅与邻居通信。在时变邻居图满足重复联合强连通性条件下，该算法可确保指数收敛至公共解，为唯一解情形推导出最坏情况下的收敛速率上界，并扩展至跟踪时变的 $A$ 和 $b$ 以及计算最小二乘解。

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.

研究动机与目标

• 设计一种分布式算法，使多个代理仅通过本地通信和部分行信息即可求解线性系统 $Ax = b$。
• 在时变有向网络中，建立代理实现对公共解指数收敛的必要与充分条件。
• 将算法扩展至当 $A$ 和 $b$ 随时间缓慢变化时，能够跟踪解。
• 将算法调整为在 $Ax = b$ 无精确解时，用于计算最小二乘解。
• 分析收敛速率与可扩展性，尤其在异步与延迟通信环境下的表现。

提出的方法

• 每个代理维护解向量的本地估计，并通过时变有向通信图 $\mathbb{N}(t)$ 接收邻居的估计值进行更新。
• 更新规则为邻居估计值的递归线性组合，旨在使所有代理趋同于 $Ax = b$ 的公共解。
• 该算法利用邻居图的结构，并假设存在重复联合强连通性以确保收敛。
• 对于最小二乘解，代理使用扩展状态向量 $\bar{x}_i$，并将其投影到由 $A_i^T A_i$ 与图拉普拉斯项组合而成的矩阵的零空间。
• 该方法在无需更新时间同步且无通信延迟的条件下仍能运行，仅需较弱的图连通性假设。
• 基于更新矩阵的谱性质与联合连通性参数 $\rho$，推导出最坏情况下的几何收敛速率上界。

实验结果

研究问题

• RQ1 时变通信图 $\mathbb{N}(t)$ 需满足何种条件，才能保证对 $Ax = b$ 的解实现指数收敛？
• RQ2 收敛速率如何依赖于网络拓扑结构与矩阵 $A$？
• RQ3 当 $A$ 和 $b$ 随时间缓慢变化时，该算法能否实现解的跟踪？其跟踪误差有何上界？
• RQ4 当 $Ax = b$ 无精确解时，如何修改该算法以计算最小二乘解？
• RQ5 通信延迟与异步更新对收敛性有何影响？是否仍能保证指数收敛？

主要发现

• 当且仅当邻居图序列 $\mathbb{N}(t)$ 重复联合强连通时，$Ax = b$ 的解可实现指数收敛至公共解。
• 对于唯一解情形，推导出最坏情况下的几何收敛速率上界，其依赖于联合连通性参数 $\rho$ 与系统的谱性质。
• 当 $A$ 和 $b$ 缓慢变化时，该算法可实现解的跟踪，且跟踪误差受 $A$ 和 $b$ 变化速率的有界控制。
• 即使在无通信延迟的异步更新条件下，该算法仍保持指数收敛。
• 该算法的修改版本可实现分布式最小二乘解计算，即使 $Ax = b$ 无精确解亦成立。
• 收敛速率上界是紧致的，依赖于 $A$ 的条件数与网络的连通性结构，其中 $\rho$ 量化了联合连通性的强度。