DC Proximal Newton for Non-Convex Optimization Problems

TL;DR

This paper presents a new proximal Newton algorithm designed for non-convex learning problems involving difference of convex functions, demonstrating improved efficiency and applicability in high-dimensional settings.

Contribution

A novel proximal Newton method capable of handling non-convex loss and regularizer functions within the DC framework, with theoretical convergence guarantees.

Findings

More efficient than existing methods for convex loss and non-convex regularizer problems.

Effective in high-dimensional transductive learning scenarios.

Proven to converge to stationary points of the DC objective.

Abstract

We introduce a novel algorithm for solving learning problems where both the loss function and the regularizer are non-convex but belong to the class of difference of convex (DC) functions. Our contribution is a new general purpose proximal Newton algorithm that is able to deal with such a situation. The algorithm consists in obtaining a descent direction from an approximation of the loss function and then in performing a line search to ensure sufficient descent. A theoretical analysis is provided showing that the iterates of the proposed algorithm {admit} as limit points stationary points of the DC objective function. Numerical experiments show that our approach is more efficient than current state of the art for a problem with a convex loss functions and non-convex regularizer. We have also illustrated the benefit of our algorithm in high-dimensional transductive learning problem where both loss function and regularizers are non-convex.