A General Family of Penalties for Combining Differing Types of Penalties in Generalized Structured Models

Penalized estimation has become an established tool for regularization and model selection in regression models. A variety of penalties with specific features are available and effective algorithms for specific penalties have been proposed. But not much is available to fit models that call for a combination of different penalties. When modeling rent data, which will be considered as an example, various types of predictors call for a combination of a Ridge, a grouped Lasso and a Lasso-type penalty within one model. Algorithms that can deal with such problems, are in demand. We propose to approximate penalties that are (semi-)norms of scalar linear transformations of the coefficient vector in generalized structured models. The penalty is very general such that the Lasso, the fused Lasso, the Ridge, the smoothly clipped absolute deviation penalty (SCAD), the elastic net and many more penalties are embedded. The approximation allows to combine all these penalties within one model. The computation is based on conventional penalized iteratively re-weighted least squares (PIRLS) algorithms and hence, easy to implement. Moreover, new penalties can be incorporated quickly. The approach is also extended to penalties with vector based arguments; that is, to penalties with norms of linear transformations of the coefficient vector. Some illustrative examples and the model for the Munich rent data show promising results