Randomized Methods for Linear Constraints: Convergence Rates and Conditioning

TL;DR

This paper analyzes randomized algorithms for solving linear systems and inequalities, providing convergence rate bounds based on condition numbers and exploring their relation to problem ill-posedness.

Contribution

It extends randomized iterative methods by establishing convergence bounds tied to linear algebraic condition numbers and discusses their generalizations.

Findings

Convergence rates depend on problem condition numbers.

Bounds relate to distances from ill-posedness.

Results apply to convex systems under metric regularity.

Abstract

We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of Strohmer and Vershynin for systems of linear equations, we show that, under appropriate probability distributions, the linear rates of convergence (in expectation) can be bounded in terms of natural linear-algebraic condition numbers for the problems. We relate these condition measures to distances to ill-posedness, and discuss generalizations to convex systems under metric regularity assumptions.