Accelerated gradient methods for nonconvex nonlinear and stochastic programming

In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the be...

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Veröffentlicht in:Mathematical programming Jg. 156; H. 1-2; S. 59 - 99
Hauptverfasser: Ghadimi, Saeed, Lan, Guanghui
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016
Springer Nature B.V
Schlagworte:
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computer programming
Convergence
Convex analysis
Full Length Paper
Machine learning
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Nonlinear programming
Numerical Analysis
Optimization
Policies
Programming
Stochastic models
Stochasticity
Studies
Theoretical
United States > US
68Q25
Nonconvex optimization
Accelerated gradient
90C25
90C15
62L20
Stochastic programming
Complexity
ISSN:0025-5610, 1436-4646
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung:In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-015-0871-8