Inference for $L_2$-Boosting
This work addresses the need for reliable inference in high-dimensional additive regression models using boosting, which is incremental as it adapts existing post-selection methods to handle iterative selection in boosting algorithms.
The authors tackled the problem of statistical inference for component-wise functional gradient descent (CFGD), also known as L2-Boosting, by proposing a framework for tests and confidence intervals for model components selected by this algorithm, with applications to an additive model for apartment sales prices and simulation studies showing its properties.
We propose a statistical inference framework for the component-wise functional gradient descent algorithm (CFGD) under normality assumption for model errors, also known as $L_2$-Boosting. The CFGD is one of the most versatile tools to analyze data, because it scales well to high-dimensional data sets, allows for a very flexible definition of additive regression models and incorporates inbuilt variable selection. Due to the variable selection, we build on recent proposals for post-selection inference. However, the iterative nature of component-wise boosting, which can repeatedly select the same component to update, necessitates adaptations and extensions to existing approaches. We propose tests and confidence intervals for linear, grouped and penalized additive model components selected by $L_2$-Boosting. Our concepts also transfer to slow-learning algorithms more generally, and to other selection techniques which restrict the response space to more complex sets than polyhedra. We apply our framework to an additive model for sales prices of residential apartments and investigate the properties of our concepts in simulation studies.