A Convergence Condition of Stochastic Approximation for Linear Equations

Abstract

Stochastic approximation is an iterative algorithm for solving an unknown equation using noisy observation data. In this paper, we revisit a convergence condition of stochastic approximation for a linear equation, where the noise is assumed to be a sequence whose time average converges to zero. In this case, it is usually assumed that the noisy coefficient matrix of the equation is symmetric, while it is not assumed in this paper. Instead, we slightly strengthen the noise convergence and show that the stochastic approximation gives the exact solution of the equation under this novel condition. The proposed condition is useful for establishing a multi-parameter stochastic approximation.