Asynchronous Approximation of a Single Component of the Solution to a Linear System

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 is an element of R-n. This can equivalently be formulated as 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 motivated by frameworks, such as Apache Spark, 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 rho(vertical bar G vertical bar) < 1, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds that 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 that 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 that computes the global solution vector x. Our algorithm obtains an epsilon parallel to x parallel to(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 = parallel to G parallel to(2)) = O(1), where parallel to G parallel to(2) is the induced matrix operator 2-norm.