On the Worst-Case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions
de Klerk,Etienne Glineur,Francois Taylor,Adrien
de Klerk,Etienne
Glineur,Francois
Taylor,Adrien
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 extend the result to 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 proof is computer-assisted, and relies on the resolution 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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2016-06-30
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Cornell University Library
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de Klerk, E, Glineur, F & Taylor, A 2016 'On the Worst-Case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions' arXiv, vol. arXiv:1606.09365, Cornell University Library, Itacha. < http://arxiv.org/abs/1606.09365 >