On the Convergence Analysis of the Optimized Gradient Method

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This paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed the optimized gradient method for this problem and showed that it has a worst-case convergence bound for the cost function decrease that is twice as small as that of Nesterov's fast gradient method, yet has a similarly efficient practical implementation. Drori showed recently that the optimized gradient method has optimal complexity for the cost function decrease over the general class of first-order methods. This optimality makes it important to study fully the convergence properties of the optimized gradient method. The previous worst-case convergence bound for the optimized gradient method was derived for only the last iterate of a secondary sequence. This paper provides an analytic convergence bound for the primary sequence generated by the optimized gradient method. We then discuss additional convergence properties of the optimized gradient method, including the interesting fact that the optimized gradient method has two types of worst-case functions: a piecewise affine-quadratic function and a quadratic function. These results help complete the theory of an optimal first-order method for smooth convex minimization.
Publisher
SPRINGER/PLENUM PUBLISHERS
Issue Date
2017-01
Language
English
Article Type
Article
Keywords

MINIMIZATION

Citation

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, v.172, no.1, pp.187 - 205

ISSN
0022-3239
DOI
10.1007/s10957-016-1018-7
URI
http://hdl.handle.net/10203/245448
Appears in Collection
MA-Journal Papers(저널논문)
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