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We investigate the asymptotic behaviour of gradient boosting algorithms whenthe learning rate converges to zero and the number of iterations is rescaledaccordingly. We mostly consider L2-boosting for regression with linear baselearner as studied in B{\"u}hlmann and Yu (2003) and analyze also a stochasticversion of the model where subsampling is used at each step (Friedman 2002). Weprove a deterministic limit in the vanishing learning rate asymptotic andcharacterize the limit as the unique solution of a linear differential equationin an infinite dimensional function space. Besides, the training and test errorof the limiting procedure are thoroughly analyzed. We finally illustrate anddiscuss our result on a simple numerical experiment where the linearL2-boosting operator is interpreted as a smoothed projection and time isrelated to its number of degrees of freedom.

Behavior of linear L2-boosting algorithms in the vanishing learning rate asymptotic