  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{\"u}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.
