线性方程组解的单分量异步逼近
Asynchronous Approximation of a Single Component of the Solution to a Linear System
摘要：
我们提出了一个分布式异步算法近似的一个单一的组成部分的解决方案的线性方程组$Ax = b$，其中$A$是一个正定的实矩阵，和$b \in \mathbb{R}^n$。这等效于对于某些$G$和$z$求解$x = Gx + z$中的$x_i$，使得$G$的光谱半径小于1。我们的算法依赖于Neumann系列表征的组件$x_i$，是基于残差更新。我们分析我们的算法的云计算模型的上下文中，在该模型中，计算被分割成小的更新任务，由小型处理器共享访问分布式文件系统。我们证明了一个强大的渐近收敛的结果时，谱半径$ρ(|G|) < 1$，无论精确的顺序和频率的更新任务的执行。我们提供的收敛速度的界限，这取决于更新任务执行的顺序，分析确定性更新规则，通过计数加权随机游走，以及概率更新规则，通过浓度界限。概率分析需要分析随机矩阵的乘积，这些随机矩阵是从时间和路径相关的分布中得出的。我们特别考虑$n$很大，而$G$很稀疏的设置，例如，每一行至多具有$d$非零条目。这是由其中$G$是从底层图的边结构导出的应用激发的。我们的研究结果证明，如果图的局部邻域不增长太快，作为$n$的函数，我们的算法可以提供显着降低计算成本，而不是任何算法计算的全局解向量$x$。我们的算法获得了$x_i$的$ε\|x\|_2$添加剂近似在常数时间相对于矩阵的大小时，最大行稀疏$d = O(1)$和$1/(1-\|G\|_2) = O(1)$。
英文摘要：
We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations $Ax = b$, where $A$ is a positive definite real matrix, and $b \in \mathbb{R}^n$. This is equivalent to solving for $x_i$ in $x = Gx + z$ for some $G$ and $z$ such that the spectral radius of $G$ is less than 1. Our algorithm relies on the Neumann series characterization of the component $x_i$, and is based on residual updates. We analyze our algorithm within the context of a cloud computation model, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius $ρ(|G|) < 1$, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds which depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices which are drawn from distributions that are time and path dependent. We specifically consider the setting where $n$ is large, yet $G$ is sparse, e.g., each row has at most $d$ nonzero entries. This is motivated by applications in which $G$ is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of $n$, our algorithm can provide significant reduction in computation cost as opposed to any algorithm which computes the global solution vector $x$. Our algorithm obtains an $ε\|x\|_2$ additive approximation for $x_i$ in constant time with respect to the size of the matrix when the maximum row sparsity $d = O(1)$ and $1/(1-\|G\|_2) = O(1)$.
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