A Pliable Lasso

Abstract

We propose a generalization of the lasso that allows the model coefficients to vary as a function of a general set of some prespecified modifying variables. These modifiers might be variables such as gender, age, or time. The paradigm is quite general, with each lasso coefficient modified by a sparse linear function of the modifying variables Z. The model is estimated in a hierarchical fashion to control the degrees of freedom and avoid overfitting. The modifying variables may be observed, observed only in the training set, or unobserved overall. There are connections of our proposal to varying coefficient models and high-dimensional interaction models. We present a computationally efficient algorithm for its optimization, with exact screening rules to facilitate application to large numbers of predictors. The method is illustrated on a number of different simulated and real examples. Supplementary materials for this article are available online.

Keywords:

• Interactions
• Linear regression
• Regularization

Acknowledgments

We would like to thank Jacob Bien and two reviewers for helpful comments.

Additional information

Funding

Robert Tibshirani was supported by NIH grant 5R01 EB001988-16 and NSF grant 19 DMS1208164.