Novel DCA based algorithms for a special class of nonconvex problems with application in machine learning

•We address the problem of minimizing the sum of a nonconvex, differentiable function and composite functions by DC (Difference of Convex functions) programming and DCA (DC Algorithm).•We first develop a standard DCA scheme especially dealing with the very specific structure of this problem.•Further...

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Vydáno v:	Applied mathematics and computation Ročník 409; s. 125904
Hlavní autoři:	Le Thi, Hoai An, Le, Hoai Minh, Phan, Duy Nhat, Tran, Bach
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Inc 15.11.2021 Elsevier
Témata:	Accelerated DCA Accelerated DCA-Like Computer Science DC programming DCA DCA-Like Dimensionality reduction Sum of nonconvex function and composite functions t-distributed Stochastic Neighbor Embedding (t-SNE) Dimensionality reduction DCA Accelerated DCA Sum of nonconvex function and composite functions DC programming DCA-Like Accelerated DCA-Like t-distributed Stochastic Neighbor Embedding (t-SNE) T-SNE Accelerated DCA-like DCA-like
ISSN:	0096-3003, 1873-5649
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Shrnutí:	•We address the problem of minimizing the sum of a nonconvex, differentiable function and composite functions by DC (Difference of Convex functions) programming and DCA (DC Algorithm).•We first develop a standard DCA scheme especially dealing with the very specific structure of this problem.•Furthermore, we extend DCA to give rise to the so-named DCA-Like, which is based on a new and efficient way to approximate the DC objective function without knowing a DC decomposition.•We further improve DCA based algorithms by incorporating the Nesterov’s acceleration technique into them.•Finally, we investigate the proposed algorithms for an important problem in machine learning: the t-distributed stochastic neighbor embedding. We address the problem of minimizing the sum of a nonconvex, differentiable function and composite functions by DC (Difference of Convex functions) programming and DCA (DC Algorithm), powerful tools of nonconvex optimization. The main idea of DCA relies on DC decompositions of the objective function, it consists in approximating a DC (nonconvex) program by a sequence of convex ones. We first develop a standard DCA scheme especially dealing with the very specific structure of this problem. Furthermore, we extend DCA to give rise to the so-named DCA-Like, which is based on a new and efficient way to approximate the DC objective function without knowing a DC decomposition. We further improve DCA based algorithms by incorporating the Nesterov’s acceleration technique into them. The convergence properties and the convergence rate under Kurdyka-Łojasiewicz assumption of extended DCAs are rigorously studied. We prove that DCA-Like and the accelerated versions subsequently converge from every initial point to a critical point of the considered problem. Finally, we investigate the proposed algorithms for an important problem in machine learning: the t-distributed stochastic neighbor embedding. Numerical experiments on several benchmark datasets illustrate the efficiency of our algorithms.
ISSN:	0096-3003 1873-5649
DOI:	10.1016/j.amc.2020.125904