# A proximal-proximal majorization-minimization algorithm for nonconvex tuning-free robust regression problems

arXiv: https://arxiv.org/abs/2106.13683 · HTML: https://mlsanity.com/paper/2106.13683/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 43 references that are in the map.
- First posted on arXiv 25 Jun 2021.

## Builds on (most cited first, 8 of 8)

- [A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems](https://mlsanity.com/paper/1607.05428/) (2016)
- [Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs](https://mlsanity.com/paper/1409.8147/) (2014)
- [Composite Difference-Max Programs for Modern Statistical Estimation Problems](https://mlsanity.com/paper/1803.00205/) (2018)
- [DC approximation approaches for sparse optimization](https://mlsanity.com/paper/1407.0286/) (2014)
- [An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming](https://mlsanity.com/paper/1903.09546/) (2019)
- [The flare Package for High Dimensional Linear Regression and Precision Matrix Estimation in R](https://mlsanity.com/paper/2006.15419/) (2020)
- [On efficiently solving the subproblems of a level-set method for fused lasso problems](https://mlsanity.com/paper/1706.08732/) (2017)
- [A sparse semismooth Newton based proximal majorization-minimization algorithm for nonconvex square-root-loss regression problems](https://mlsanity.com/paper/1903.11460/) (2019)

## Abstract (arXiv)

In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex tuning-free robust regression problems. The basic idea is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). We must emphasize that the main difficulty in the design of the algorithm lies in how to overcome the singular difficulty of the inner subproblem. Furthermore, we also prove that the PPMM algorithm converges to a d-stationary point. Due to the Kurdyka-Lojasiewicz (KL) property of the problem, we present the convergence rate of the PPMM algorithm. Numerical experiments demonstrate that our proposed algorithm outperforms the existing state-of-the-art algorithms.