Randomized Iterative Methods for Linear Systems

TL;DR

This paper introduces a unified randomized iterative framework for solving linear systems, encompassing many existing algorithms and enabling new variants with proven exponential convergence.

Contribution

It presents a novel, versatile randomized method with multiple interpretations that generalizes and unifies several known algorithms for linear systems.

Findings

Proves exponential convergence of the expected error norm.

Derives exact formulas for the evolution of expected iterates.

Provides lower bounds on convergence rates.

Abstract

We develop a novel, fundamental and surprisingly simple randomized iterative method for solving consistent linear systems. Our method has six different but equivalent interpretations: sketch-and-project, constrain-and-approximate, random intersect, random linear solve, random update and random fixed point. By varying its two parameters$-$a positive definite matrix (defining geometry), and a random matrix (sampled in an independently and identically distributed fashion in each iteration)$-$we recover a comprehensive array of well-known algorithms as special cases, including the randomized Kaczmarz method, randomized Newton method, randomized coordinate descent method and random Gaussian pursuit. We naturally also obtain variants of all these methods using blocks and importance sampling. However, our method allows for a much wider selection of these two parameters, which leads to a number of new specific methods. We prove exponential convergence of the expected norm of the error in a single theorem, from which existing complexity results for known variants can be obtained. However, we also give an exact formula for the evolution of the expected iterates, which allows us to give lower bounds on the convergence rate.