Analytic analysis of the worst-case complexity of the gradient method with exact line search and the Polyak stepsize

TL;DR

This paper provides a new analytical approach to determine the worst-case complexity of gradient methods with exact line search and Polyak stepsize, revealing their convergence behavior and zigzag patterns.

Contribution

It introduces a novel analytic framework for worst-case complexity analysis of gradient methods with specific stepsizes, previously only accessible via computer-assisted proofs.

Findings

Gradient method with Polyak stepsize zigzags in a two-dimensional subspace.

The analysis characterizes the asymptotic behavior of the gradient methods.

The approach unifies the analysis of exact line search and Polyak stepsize methods.

Abstract

We give a novel analytic analysis of the worst-case complexity of the gradient method with exact line search and the Polyak stepsize, respectively, which previously could only be established by computer-assisted proof. Our analysis is based on studying the linear convergence of a family of gradient methods, whose stepsizes include the one determined by exact line search and the Polyak stepsize as special instances. The asymptotic behavior of the considered family is also investigated which shows that the gradient method with the Polyak stepsize will zigzag in a two-dimensional subspace spanned by the two eigenvectors corresponding to the largest and smallest eigenvalues of the Hessian.