Abstract
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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This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The first author is a F.R.S.-FNRS Research Fellow. The research of the third author was partly supported by the grant Action de recherche concertée ARC 04/09-315 from the Direction de la recherche scientifique—Communauté française de Belgique. The third author also acknowledges the support from Laboratory of Structural Methods of Data Analysis in Predictive Modelling, through the RF government grant 11.G34.31.0073. The scientific responsibility rests with its authors.
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Devolder, O., Glineur, F. & Nesterov, Y. First-order methods of smooth convex optimization with inexact oracle. Math. Program. 146, 37–75 (2014). https://doi.org/10.1007/s10107-013-0677-5
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DOI: https://doi.org/10.1007/s10107-013-0677-5
Keywords
- Smooth convex optimization
- First-order methods
- Inexact oracle
- Gradient methods
- Fast gradient methods
- Complexity bounds