Behavior of linear L2-boosting algorithms in the vanishing learning rate asymptotic
This work provides a theoretical characterization of the asymptotic behavior of L2-boosting, which is significant for researchers and practitioners seeking a deeper understanding of boosting algorithms' convergence properties.
This paper investigates the asymptotic behavior of L2-boosting algorithms with linear base learners as the learning rate approaches zero. It proves a deterministic limit for the procedure, characterized as the unique solution of a linear differential equation in an infinite-dimensional function space, and analyzes the training and test errors of this limiting process.
We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly consider L2-boosting for regression with linear base learner as studied in B{ü}hlmann and Yu (2003) and analyze also a stochastic version of the model where subsampling is used at each step (Friedman 2002). We prove a deterministic limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the training and test error of the limiting procedure are thoroughly analyzed. We finally illustrate and discuss our result on a simple numerical experiment where the linear L2-boosting operator is interpreted as a smoothed projection and time is related to its number of degrees of freedom.