On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions
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].
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Cite
@article{arxiv.1606.09365,
title = {On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions},
author = {Etienne de Klerk and François Glineur and Adrien B. Taylor},
journal= {arXiv preprint arXiv:1606.09365},
year = {2016}
}
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11pages