DC Proximal Newton for Non-Convex Optimization Problems
This work addresses optimization challenges in machine learning for non-convex problems, offering a method that is incremental but improves efficiency in specific scenarios like high-dimensional transductive learning.
The authors tackled the problem of optimizing non-convex functions that are differences of convex functions by introducing a novel proximal Newton algorithm, which was shown to be more efficient than state-of-the-art methods in numerical experiments, particularly for problems with convex loss and non-convex regularizers.
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.