Lasso Screening Rules via Dual Polytope Projection

TL;DR

This paper introduces a new screening rule based on dual polytope projections that efficiently identifies inactive predictors in large-scale Lasso problems, significantly reducing computational complexity.

Contribution

It proposes an exact screening rule for Lasso and extends it to group Lasso, improving predictor elimination efficiency over existing methods.

Findings

The screening rule effectively identifies inactive predictors in synthetic and real datasets.

It outperforms existing screening rules in terms of accuracy and efficiency.

The method is applicable to large-scale high-dimensional problems.

Abstract

Lasso is a widely used regression technique to find sparse representations. When the dimension of the feature space and the number of samples are extremely large, solving the Lasso problem remains challenging. To improve the efficiency of solving large-scale Lasso problems, El Ghaoui and his colleagues have proposed the SAFE rules which are able to quickly identify the inactive predictors, i.e., predictors that have $0$ components in the solution vector. Then, the inactive predictors or features can be removed from the optimization problem to reduce its scale. By transforming the standard Lasso to its dual form, it can be shown that the inactive predictors include the set of inactive constraints on the optimal dual solution. In this paper, we propose an efficient and effective screening rule via Dual Polytope Projections (DPP), which is mainly based on the uniqueness and nonexpansiveness of the optimal dual solution due to the fact that the feasible set in the dual space is a convex and closed polytope. Moreover, we show that our screening rule can be extended to identify inactive groups in group Lasso. To the best of our knowledge, there is currently no "exact" screening rule for group Lasso. We have evaluated our screening rule using synthetic and real data sets. Results show that our rule is more effective in identifying inactive predictors than existing state-of-the-art screening rules for Lasso.