Gert Lanckriet, Bharath K. Sriperumbudur
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), their 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 Zangwills 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 Zangwills 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.