Adaptive step-length selection in gradient boosting for Gaussian location and scale models

Tuning of model-based boosting algorithms relies mainly on the number of iterations, while the step-length is fixed at a predefined value. For complex models with several predictors such as Generalized additive models for location, scale and shape (GAMLSS), imbalanced updates of predictors, where some distribution parameters are updated more frequently than others, can be a problem that prevents some submodels to be appropriately fitted within a limited number of boosting iterations. We propose an approach using adaptive step-length (ASL) determination within a non-cyclical boosting algorithm for Gaussian location and scale models, as an important special case of the wider class of GAMLSS, to prevent such imbalance. Moreover, we discuss properties of the ASL and derive a semi-analytical form of the ASL that avoids manual selection of the search interval and numerical optimization to find the optimal step-length, and consequently improves computational efficiency. We show competitive behavior of the proposed approaches compared to penalized maximum likelihood and boosting with a fixed step-length for Gaussian location and scale models in two simulations and two applications, in particular for cases of large variance and/or more variables than observations. In addition, the underlying concept of the ASL is also applicable to the whole GAMLSS framework and to other models with more than one predictor like zero-inflated count models, and brings up insights into the choice of the reasonable defaults for the step-length in the simpler special case of (Gaussian) additive models.