Parallel ADMM Algorithm with Gaussian Back Substitution for High-Dimensional Quantile Regression and Classification

TL;DR

This paper introduces a Gaussian Back-Substitution strategy to enhance the convergence and efficiency of parallel ADMM algorithms for high-dimensional quantile regression and classification, including novel models.

Contribution

It proposes a simple correction step that guarantees linear convergence of parallel ADMM algorithms for quantile problems, extending to new classification models.

Findings

The modified P-ADMM achieves reliable convergence.

Numerical simulations show high efficiency.

The method effectively handles high-dimensional data.

Abstract

In the field of high-dimensional data analysis, modeling methods based on quantile loss function are highly regarded due to their ability to provide a comprehensive statistical perspective and effective handling of heterogeneous data. In recent years, many studies have focused on using the parallel alternating direction method of multipliers (P-ADMM) to solve high-dimensional quantile regression and classification problems. One efficient strategy is to reformulate the quantile loss function by introducing slack variables. However, this reformulation introduces a theoretical challenge: even when the regularization term is convex, the convergence of the algorithm cannot be guaranteed. To address this challenge, this paper proposes the Gaussian Back-Substitution strategy, which requires only a simple and effective correction step that can be easily integrated into existing parallel algorithm frameworks, achieving a linear convergence rate. Furthermore, this paper extends the parallel algorithm to handle some novel quantile loss classification models. Numerical simulations demonstrate that the proposed modified P-ADMM algorithm exhibits excellent performance in terms of reliability and efficiency.