Bach Tran

Hoai An Le Thi, Hoai Minh Le, Duy Nhat Phan et al.

We consider the large sum of DC (Difference of Convex) functions minimization problem which appear in several different areas, especially in stochastic optimization and machine learning. Two DCA (DC Algorithm) based algorithms are proposed: stochastic DCA and inexact stochastic DCA. We prove that the convergence of both algorithms to a critical point is guaranteed with probability one. Furthermore, we develop our stochastic DCA for solving an important problem in multi-task learning, namely group variables selection in multi class logistic regression. The corresponding stochastic DCA is very inexpensive, all computations are explicit. Numerical experiments on several benchmark datasets and synthetic datasets illustrate the efficiency of our algorithms and their superiority over existing methods, with respect to classification accuracy, sparsity of solution as well as running time.

Hoai An Le Thi, Hoai Minh Le, Duy Nhat Phan et al.

In this paper, we present two variants of DCA (Different of Convex functions Algorithm) to solve the constrained sum of differentiable function and composite functions minimization problem, with the aim of increasing the convergence speed of DCA. In the first variant, DCA-Like, we introduce a new technique to iteratively modify the decomposition of the objective function. This successive decomposition could lead to a better majorization and consequently a better convergence speed than the basic DCA. We then incorporate the Nesterov's acceleration technique into DCA-Like to give rise to the second variant, named Accelerated DCA-Like. The convergence properties and the convergence rate under Kudyka-Lojasiewicz assumption of both variants are rigorously studied. As an application, we investigate our algorithms for the t-distributed stochastic neighbor embedding. Numerical experiments on several benchmark datasets illustrate the efficiency of our algorithms.