The Boosted DC Algorithm for nonsmooth functions

TL;DR

This paper extends the Boosted DC Algorithm (BDCA) to nonsmooth functions, demonstrating its convergence and superior performance in data science applications like clustering and multidimensional scaling.

Contribution

It generalizes BDCA to nonsmooth DC functions and introduces flexible step size choices, improving convergence and efficiency.

Findings

BDCA outperforms DCA in speed by up to 16 times in clustering.

BDCA is three times faster than DCA in multidimensional scaling.

Theoretical convergence is established under the Kurdyka-Lojasiewicz property.

Abstract

The Boosted Difference of Convex functions Algorithm (BDCA) was recently proposed for minimizing smooth difference of convex (DC) functions. BDCA accelerates the convergence of the classical Difference of Convex functions Algorithm (DCA) thanks to an additional line search step. The purpose of this paper is twofold. Firstly, to show that this scheme can be generalized and successfully applied to certain types of nonsmooth DC functions, namely, those that can be expressed as the difference of a smooth function and a possibly nonsmooth one. Secondly, to show that there is complete freedom in the choice of the trial step size for the line search, which is something that can further improve its performance. We prove that any limit point of the BDCA iterative sequence is a critical point of the problem under consideration, and that the corresponding objective value is monotonically decreasing and convergent. The global convergence and convergent rate of the iterations are obtained under the Kurdyka-Lojasiewicz property. Applications and numerical experiments for two problems in data science are presented, demonstrating that BDCA outperforms DCA. Specifically, for the Minimum Sum-of-Squares Clustering problem, BDCA was on average sixteen times faster than DCA, and for the Multidimensional Scaling problem, BDCA was three times faster than DCA.