First-order methods of smooth convex optimization with inexact oracle

We introduce the notion of inexact first-order oracle and analyze the behavior of several first-order methods of smooth convex optimization used with such an oracle. This notion of inexact oracle naturally appears in the context of smoothing techniques, Moreau–Yosida regularization, Augmented Lagran...

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Bibliographic Details
Published in:Mathematical programming Vol. 146; no. 1-2; pp. 37 - 75
Main Authors: Devolder, Olivier, Glineur, François, Nesterov, Yurii
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014
Springer Nature B.V
Subjects:
Accuracy
Analysis
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Complexity theory
Convex analysis
Estimates
Full Length Paper
Functions (mathematics)
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Permissible error
Regularization
Smoothing
Studies
Theoretical
Gradient methods
Fast gradient methods
90C25
90C06
Inexact oracle
Complexity bounds
90C60
First-order methods
Smooth convex optimization
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We introduce the notion of inexact first-order oracle and analyze the behavior of several first-order methods of smooth convex optimization used with such an oracle. This notion of inexact oracle naturally appears in the context of smoothing techniques, Moreau–Yosida regularization, Augmented Lagrangians and many other situations. We derive complexity estimates for primal, dual and fast gradient methods, and study in particular their dependence on the accuracy of the oracle and the desired accuracy of the objective function. We observe that the superiority of fast gradient methods over the classical ones is no longer absolute when an inexact oracle is used. We prove that, contrary to simple gradient schemes, fast gradient methods must necessarily suffer from error accumulation. Finally, we show that the notion of inexact oracle allows the application of first-order methods of smooth convex optimization to solve non-smooth or weakly smooth convex problems.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-013-0677-5