On the convergence of the concave-convex procedure

Bharath Kumar Sriperumbudur, Gert R.G. Lanckriet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

236 Scopus citations

Abstract

The concave-convex procedure (CCCP) is a majorization-minimization algorithm that solves d.c. (difference of convex functions) programs as a sequence of convex programs. In machine learning, CCCP is extensively used in many learning algorithms like sparse support vector machines (SVMs), transductive SVMs, sparse principal component analysis, etc. Though widely used in many applications, the convergence behavior of CCCP has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper, however, we believe the analysis is not complete. Although the convergence of CCCP can be derived from the convergence of the d.c. algorithm (DCA), its proof is more specialized and technical than actually required for the specific case of CCCP. In this paper, we follow a different reasoning and show how Zangwill's global convergence theory of iterative algorithms provides a natural framework to prove the convergence of CCCP, allowing a more elegant and simple proof. This underlines Zangwill's theory as a powerful and general framework to deal with the convergence issues of iterative algorithms, after also being used to prove the convergence of algorithms like expectation- maximization, generalized alternating minimization, etc. In this paper, we provide a rigorous analysis of the convergence of CCCP by addressing these questions: (i) When does CCCP find a local minimum or a stationary point of the d.c. program under consideration? (ii) When does the sequence generated by CCCP converge? We also present an open problem on the issue of local convergence of CCCP.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference
Pages1759-1767
Number of pages9
StatePublished - Dec 1 2009
Event23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 - Vancouver, BC, Canada
Duration: Dec 7 2009Dec 10 2009

Publication series

NameAdvances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference

Other

Other23rd Annual Conference on Neural Information Processing Systems, NIPS 2009
CountryCanada
CityVancouver, BC
Period12/7/0912/10/09

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All Science Journal Classification (ASJC) codes

  • Information Systems

Cite this

Sriperumbudur, B. K., & Lanckriet, G. R. G. (2009). On the convergence of the concave-convex procedure. In Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference (pp. 1759-1767). (Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference).