First-order methods with inexact oracle: the strongly convex case
The goal of this paper is to study the effect of inexact first-order information on the first-order methods designed for smooth strongly convex optimization problems. It can be seen as a generalization to the strongly convex case of our previous paper . We introduce the notion of (δ,L,μ)-oracle, that can be seen as an extension of the (δ,L)-oracle (previously introduced in ), taking into account strong convexity. We consider different examples of (δ,L,μ)-oracle: strongly convex function with first-order information computed at a shifted point, strongly convex function with approximate gradient and strongly convex max-function with inexact resolution of subproblems. The core of this paper is devoted to the behavior analysis of three first-order methods, respectively the primal, the dual and the fast gradient method, when used with a (δ, L, μ)- oracle. As in the smooth convex case (studied in ), we obtain that the simple gradient methods can be seen as robust but relatively slow, whereas the fast gradient method is faster but more sensitive to oracle errors. However, the strong convexity leads to much faster convergence rates (linear instead of sublinear) for every method and to a reduced sensitivity with respect to oracle errors. We also prove that the notion of (δ, L, μ)-oracle can be used in order to model exact first-order information but for functions with weaker level of smoothness and different level of convexity. This observation allows us to apply methods, originally designed for smooth strongly convex function, to weakly smooth uniformly convex functions and to derive corresponding performance guarantees.
|Date of creation:||17 May 2013|
|Contact details of provider:|| Postal: Voie du Roman Pays 34, 1348 Louvain-la-Neuve (Belgium)|
Fax: +32 10474304
Web page: http://www.uclouvain.be/core
More information through EDIRC
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Fleurbaey,Marc & Maniquet,FranÃ§ois, 2011.
"A Theory of Fairness and Social Welfare,"
Cambridge University Press, number 9780521715348, March.
- Fleurbaey,Marc & Maniquet,FranÃ§ois, 2011. "A Theory of Fairness and Social Welfare," Cambridge Books, Cambridge University Press, number 9780521887427, March.
- DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," CORE Discussion Papers 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Duranton, Gilles & Martin, Philippe & Mayer, Thierry & Mayneris, Florian, 2010. "The Economics of Clusters: Lessons from the French Experience," OUP Catalogue, Oxford University Press, number 9780199592203. Full references (including those not matched with items on IDEAS)
When requesting a correction, please mention this item's handle: RePEc:cor:louvco:2013016. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Alain GILLIS)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.