A Unified Algorithm for Penalized Convolution Smoothed Quantile Regression

TL;DR

This paper introduces a unified, efficient algorithm for penalized convolution smoothed quantile regression that handles various penalties, improving computational speed and accuracy in high-dimensional data analysis.

Contribution

It proposes a novel, unified algorithm for penalized convolution smoothed QR with multiple penalties, along with an R package for practical implementation.

Findings

Outperforms existing methods in speed and accuracy

Demonstrates effectiveness on real-world data

Provides a versatile tool for high-dimensional quantile regression

Abstract

Penalized quantile regression (QR) is widely used for studying the relationship between a response variable and a set of predictors under data heterogeneity in high-dimensional settings. Compared to penalized least squares, scalable algorithms for fitting penalized QR are lacking due to the non-differentiable piecewise linear loss function. To overcome the lack of smoothness, a recently proposed convolution-type smoothed method brings an interesting tradeoff between statistical accuracy and computational efficiency for both standard and penalized quantile regressions. In this paper, we propose a unified algorithm for fitting penalized convolution smoothed quantile regression with various commonly used convex penalties, accompanied by an R-language package conquer available from the Comprehensive R Archive Network. We perform extensive numerical studies to demonstrate the superior performance of the proposed algorithm over existing methods in both statistical and computational aspects. We further exemplify the proposed algorithm by fitting a fused lasso additive QR model on the world happiness data.