# A proximal dual semismooth Newton method for computing zero-norm penalized QR estimator

## TL;DR

This paper introduces a novel multi-stage convex relaxation method using a proximal dual semismooth Newton approach to efficiently compute high-dimensional zero-norm penalized quantile regression estimators, with theoretical guarantees and superior empirical performance.

## Contribution

It develops a new multi-stage convex relaxation algorithm with a proximal dual semismooth Newton method for zero-norm penalized QR, providing theoretical error bounds and convergence analysis.

## Key findings

- Achieves linear convergence rate under restricted strong convexity.
- Outperforms existing methods in estimation accuracy and computational efficiency.
- Demonstrates effectiveness on synthetic and real datasets.

## Abstract

This paper is concerned with the computation of the high-dimensional zero-norm penalized quantile regression estimator, defined as a global minimizer of the zero-norm penalized check loss function. To seek a desirable approximation to the estimator, we reformulate this NP-hard problem as an equivalent augmented Lipschitz optimization problem, and exploit its coupled structure to propose a multi-stage convex relaxation approach (MSCRA\_PPA), each step of which solves inexactly a weighted $\ell_1$-regularized check loss minimization problem with a proximal dual semismooth Newton method. Under a restricted strong convexity condition, we provide the theoretical guarantee for the MSCRA\_PPA by establishing the error bound of each iterate to the true estimator and the rate of linear convergence in a statistical sense. Numerical comparisons on some synthetic and real data show that MSCRA\_PPA not only has comparable even better estimation performance, but also requires much less CPU time.