On a Family of Relaxed Gradient Descent Methods for Quadratic Minimization

TL;DR

This paper analyzes a family of Relaxed ll-Minimal Gradient Descent methods for quadratic optimization, proving linear convergence and demonstrating practical benefits through numerical experiments.

Contribution

It introduces a unified convergence analysis for the entire family of relaxed gradient descent methods, including Steepest Descent and Minimal Gradient, with new iteration complexity results.

Findings

Linear convergence of gradient and iterate distance in an appropriate norm

Function values decrease linearly during optimization

Relaxation improves practical performance as shown in numerical experiments

Abstract

This paper studies the convergence properties of a family of Relaxed $\ell$-Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method. Simple proofs are provided that show, in an appropriately chosen norm, the gradient and the distance of the iterates from optimality converge linearly, for all members of the family. Moreover, the function values decrease linearly, and iteration complexity results are provided. All theoretical results hold when (fixed) relaxation is employed. It is also shown that, given a fixed overhead and storage budget, every Relaxed $\ell$-Minimal Gradient Descent method can be implemented using exactly one matrix vector product. Numerical experiments are presented that illustrate the benefits of relaxation across the family.