Stochastic DCA for minimizing a large sum of DC functions with application to Multi-class Logistic Regression

TL;DR

This paper introduces stochastic and inexact stochastic DCA algorithms for efficiently minimizing large sums of DC functions, with proven convergence and demonstrated effectiveness in multi-class logistic regression tasks.

Contribution

It develops novel stochastic DCA algorithms with guaranteed convergence for large-scale DC problems and applies them to multi-task learning, showing improved performance.

Findings

Algorithms converge to critical points with probability one.

Stochastic DCA outperforms existing methods in accuracy and sparsity.

Methods are computationally inexpensive and effective on benchmark datasets.

Abstract

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.