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A LASSO FOR HIERARCHICAL INTERACTIONS.

Jacob Bien¹, Jonathan Taylor¹, Robert Tibshirani¹

¹Cornell University, Stanford University and Stanford University.

Annals of Statistics

|August 11, 2015

Summary

This summary is machine-generated.

This study introduces sparse interaction models with hierarchy constraints, focusing on practical sparsity for efficient data collection. The new method ensures interactions are only included if variables are marginally important, reducing measurement needs.

Keywords:

Regularized regression convexity hierarchical sparsity interactions lasso

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Area of Science:

Statistics
Machine Learning
Computational Biology

Background:

The Lasso method can produce sparse models but does not inherently enforce hierarchical relationships between variables.
Variable interactions are crucial in many fields, but their inclusion in models can be computationally expensive and data-intensive.
Distinguishing between parameter sparsity and practical sparsity is key for efficient data collection and analysis.

Purpose of the Study:

To develop a method for creating sparse interaction models that respect the hierarchy constraint.
To provide a theoretical understanding and practical implementation of hierarchical sparse modeling.
To reduce the number of raw variables that need to be measured for accurate predictions.

Main Methods:

Incorporating convex constraints into the Lasso algorithm to enforce hierarchy.
Developing an unbiased estimator for the degrees of freedom of the hierarchical model.
Analyzing the impact of the hierarchy constraint on model fitting and variable selection.

Main Results:

A precise characterization of the hierarchy constraint's effect on model sparsity.
Proof that the hierarchy holds with probability one.
Derivation of a bound on the degrees of freedom estimate, quantifying the "saved" fitting due to hierarchy.

Conclusions:

The proposed method effectively generates sparse interaction models that honor hierarchy.
The focus on practical sparsity offers significant advantages in terms of data collection efficiency.
The developed algorithm, implemented in the R package hierNet, provides a practical tool for researchers.