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arXiv · 1607.06764

Generalizing the optimized gradient method for smooth convex minimization

Donghwan Kim·Jeffrey A. Fessler
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Abstract

This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate $O(1/N^{1.5})$ on the decrease of the gradient norm among fixed-step first-order methods. This paper also proves that Nesterov's fast gradient method has an $O(1/N^{1.5})$ worst-case gradient norm rate but with constant larger than OGM-OG. The proof is based on the worst-case analysis called Performance Estimation Problem.

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Donghwan Kim, Jeffrey A. Fessler. 2016-07-22. Generalizing the optimized gradient method for smooth convex minimization. https://doi.org/10.1137/17m112124x

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