Parameter Estimation with the Ordered $\ell_{2}$ Regularization via an Alternating Direction Method of Multipliers

TL;DR

This paper introduces ADMM-Oℓ₂, a scalable algorithm for ordered ℓ₂ regularization that effectively handles high-dimensional, large-scale data, automatically excludes irrelevant variables, and converges quickly.

Contribution

The paper proposes ADMM-Oℓ₂, a novel method that scales ordered ℓ₂ regularization to large datasets and improves parameter estimation accuracy.

Findings

ADMM-Oℓ₂ outperforms existing methods on synthetic data.

ADMM-Oℓ₂ performs comparably on real datasets.

The method converges rapidly in high-dimensional settings.

Abstract

Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data than traditional regularization. The reason stems from the fact that the ordered regularization can reject irrelevant variables and yield an accurate estimation of the parameters. How to scale up the ordered regularization problems when facing the large-scale training data remains an unanswered question. This paper explores the problem of parameter estimation with the ordered $\ell_{2}$-regularization via Alternating Direction Method of Multipliers (ADMM), called ADMM-O$\ell_{2}$. The advantages of ADMM-O$\ell_{2}$ include (i) scaling up the ordered $\ell_{2}$ to a large-scale dataset, (ii) predicting parameters correctly by excluding irrelevant variables automatically, and (iii) having a fast convergence rate. Experiment results on both synthetic data and real data indicate that ADMM-O$\ell_{2}$ can perform better than or comparable to several state-of-the-art baselines.