Randomized iterative methods for linear systems

Robert M. Gower, Peter Richtarik

Research output: Contribution to journalArticlepeer-review

146 Scopus citations


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 independent 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.

Original languageEnglish (US)
Pages (from-to)1660-1690
Number of pages31
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
StatePublished - 2015
Externally publishedYes


  • Iterative Methods
  • Linear Systems
  • Random Pursuit
  • Randomized Coordinate Descent
  • Randomized Fixed Point
  • Randomized Kaczmarz
  • Randomized Newton
  • Stochastic Methods

ASJC Scopus subject areas

  • Analysis


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