# On Quadratic Convergence of DC Proximal Newton Algorithm for Nonconvex Sparse Learning in High Dimensions

arXiv: https://arxiv.org/abs/1706.06066 · HTML: https://mlsanity.com/paper/1706.06066/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2017.

## Built on by (most cited first, 2 of 2)

- [Best-Subset Selection in Generalized Linear Models: A Fast and Consistent Algorithm via Splicing Technique](https://mlsanity.com/paper/2308.00251/) (2023)
- [An Improved Boosted DC Algorithm for Nonsmooth Functions with Applications in Image Recovery](https://mlsanity.com/paper/2602.04237/) (2026)

## Abstract (arXiv)

We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization of sparse modeling structures/assumptions (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) quadratic convergence, and eventually obtains a sparse approximate local optimum with optimal statistical properties after only a few convex relaxations. Numerical experiments are provided to support our theory.