Sparse Partially Linear Additive Models
This work addresses model selection issues in flexible predictive modeling for scenarios with little prior feature knowledge, offering an incremental improvement over existing methods like lasso and sparse additive models.
The paper tackles the challenge of automatically selecting features and determining their linear or nonlinear effects in generalized partially linear additive models (GPLAMs) by introducing the sparse partially linear additive model (SPLAM), which integrates these selections into a convex optimization problem and demonstrates improved performance in high-dimensional settings with up to 500,000 samples and 45,000 features.
The generalized partially linear additive model (GPLAM) is a flexible and interpretable approach to building predictive models. It combines features in an additive manner, allowing each to have either a linear or nonlinear effect on the response. However, the choice of which features to treat as linear or nonlinear is typically assumed known. Thus, to make a GPLAM a viable approach in situations in which little is known $a~priori$ about the features, one must overcome two primary model selection challenges: deciding which features to include in the model and determining which of these features to treat nonlinearly. We introduce the sparse partially linear additive model (SPLAM), which combines model fitting and $both$ of these model selection challenges into a single convex optimization problem. SPLAM provides a bridge between the lasso and sparse additive models. Through a statistical oracle inequality and thorough simulation, we demonstrate that SPLAM can outperform other methods across a broad spectrum of statistical regimes, including the high-dimensional ($p\gg N$) setting. We develop efficient algorithms that are applied to real data sets with half a million samples and over 45,000 features with excellent predictive performance.