Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penalties

TL;DR

This paper introduces a novel smoothing ADMM algorithm called SIAD for sparse-penalized quantile regression with non-convex penalties, achieving faster convergence and improved stability over existing methods.

Contribution

The paper proposes the SIAD algorithm with a single-loop smoothing approach and increasing penalty, along with convergence analysis and empirical validation for sparse-penalized quantile regression.

Findings

SIAD outperforms existing methods in speed and stability.

Convergence rate of the sub-gradient bound is established as o(k^{-1/4}).

Numerical results confirm the effectiveness of SIAD.

Abstract

This paper investigates quantile regression in the presence of non-convex and non-smooth sparse penalties, such as the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD). The non-smooth and non-convex nature of these problems often leads to convergence difficulties for many algorithms. While iterative techniques like coordinate descent and local linear approximation can facilitate convergence, the process is often slow. This sluggish pace is primarily due to the need to run these approximation techniques until full convergence at each step, a requirement we term as a \emph{secondary convergence iteration}. To accelerate the convergence speed, we employ the alternating direction method of multipliers (ADMM) and introduce a novel single-loop smoothing ADMM algorithm with an increasing penalty parameter, named SIAD, specifically tailored for sparse-penalized quantile regression. We first delve into the convergence properties of the proposed SIAD algorithm and establish the necessary conditions for convergence. Theoretically, we confirm a convergence rate of $o\big({k^{-\frac{1}{4}}}\big)$ for the sub-gradient bound of augmented Lagrangian. Subsequently, we provide numerical results to showcase the effectiveness of the SIAD algorithm. Our findings highlight that the SIAD method outperforms existing approaches, providing a faster and more stable solution for sparse-penalized quantile regression.