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Randomization has shown catalyzing effects in linear algebra with promisingperspectives for tackling computational challenges in large-scale problems. Forsolving a system of linear equations, we demonstrate the convergence of a broadclass of algorithms that at each step pick a subset of $n$ equations at randomand update the iterate with the orthogonal projection to the subspace thoseequations represent. We identify, in this context, a specific degree-$n$polynomial that non-linearly transforms the singular values of the systemtowards equalization. This transformation to singular values and thecorresponding condition number then characterizes the expected convergence rateof iterations. As a consequence, our results specify the convergence rate ofthe stochastic gradient descent algorithm, in terms of the mini-batch size $n$,when used for solving systems of linear equations.

Solving A System Of Linear Equations By Randomized Orthogonal Projections