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Consider the classical problem of solving a general linear system ofequations $Ax=b$. It is well known that the (successively over relaxed)Gauss-Seidel scheme and many of its variants may not converge when $A$ isneither diagonally dominant nor symmetric positive definite. Can we have alinearly convergent G-S type algorithm that works for {\it any} $A$? In thispaper we answer this question affirmatively by proposing a doubly stochasticG-S algorithm that is provably linearly convergent (in the mean square errorsense) for any feasible linear system of equations. The key in the algorithmdesign is to introduce a {\it nonuniform double stochastic} scheme for pickingthe equation and the variable in each update step as well as a stepsize rule.These techniques also generalize to certain iterative alternating projectionalgorithms for solving the linear feasibility problem $A x\le b$ with anarbitrary $A$, as well as high-dimensional minimization problems for trainingover-parameterized models in machine learning. Our results demonstrate that acarefully designed randomization scheme can make an otherwise divergent G-Salgorithm converge.

A Linearly Convergent Doubly Stochastic Gauss-Seidel Algorithm for  Solving Linear Equations and A Certain Class of Over-Parameterized  Optimization Problems