Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach

Achintya Kundu 1 Francis Bach 2 Chiranjib Bhattacharyya 3
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01652149
Contributeur : Francis Bach <>
Soumis le : jeudi 30 novembre 2017 - 07:55:35
Dernière modification le : jeudi 26 avril 2018 - 10:28:58

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Achintya Kundu, Francis Bach, Chiranjib Bhattacharyya. Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach. 2017. 〈hal-01652149〉

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