Accelerated and Inexact Forward-Backward Algorithms

We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the $1/k^2$ convergence rate for the function values...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 23; no. 3; pp. 1607 - 1633
Main Authors: Villa, Silvia, Salzo, Saverio, Baldassarre, Luca, Verri, Alessandro
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects:
Algorithms
Approximation
Computing costs
Convergence
Cost analysis
Estimates
Exact solutions
Laboratories
Machine learning
Mathematical analysis
Optimization
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:We propose a convergence analysis of accelerated forward-backward splitting methods for composite function minimization, when the proximity operator is not available in closed form, and can only be computed up to a certain precision. We prove that the $1/k^2$ convergence rate for the function values can be achieved if the admissible errors are of a certain type and satisfy a sufficiently fast decay condition. Our analysis is based on the machinery of estimate sequences first introduced by Nesterov for the study of accelerated gradient descent algorithms. Furthermore, we give a global complexity analysis, taking into account the cost of computing admissible approximations of the proximal point. An experimental analysis is also presented. [PUBLICATION ABSTRACT]
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ISSN:1052-6234
1095-7189
DOI:10.1137/110844805