On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions

TL;DR

This paper precisely characterizes the worst-case convergence rate of the gradient method with exact line search on smooth strongly convex functions, using computer-assisted proofs based on semidefinite programming.

Contribution

It provides the exact worst-case convergence rate and tight complexity bounds for both noiseless and noisy gradient methods with line search, employing semidefinite programming techniques.

Findings

Worst-case convergence rate established for the gradient method with line search.

Tight complexity bounds derived for noisy gradient descent variants.

Performance estimation problems solved via semidefinite programming.

Abstract

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also give the tight worst-case complexity bound for a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proofs are computer-assisted, and rely on the resolutions of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].