Revisiting Asynchronous Linear Solvers: Provable Convergence Rate through Randomization

Asynchronous methods for solving systems of linear equations have been researched since Chazan and Mir Anker's pioneering 1969 paper. The underlying idea of asynchronous methods is to avoid processor idle time by allowing the processors to continue to make progress even if not all progress made by other processors has been communicated to them. Historically, work on asynchronous methods for solving linear equations focused on proving convergence in the limit. %How the rate of convergence %compares to the rate of convergence of the synchronous counterparts, %and how this rate scales when the number of processors increase, was %seldom studied and is still not well understood. Comparison of the asynchronous convergence rate with its synchronous counterpart and its scaling with the number of processors were seldom studied, and are still not well understood. Furthermore, the applicability of these methods was limited to restricted classes of matrices, such as diagonally dominant matrices. We propose a randomized shared-memory asynchronous method for general symmetric positive definite matrices. We rigorously analyze the convergence rate and prove that it is linear, and is close to that of the method's synchronous counterpart if the processor count is not excessive relative to the size and sparsity of the matrix. Our work presents a significant improvement in convergence analysis as well as in the applicability of asynchronous linear solvers, and suggests randomization as a key paradigm to serve as a foundation for asynchronous methods.