  Given functional data from a survival process with time-dependent covariates,
we derive a smooth convex representation for its nonparametric log-likelihood
functional and obtain its functional gradient. From this, we devise a generic
gradient boosting procedure for estimating the hazard function
nonparametrically. An illustrative implementation of the procedure using
regression trees is described to show how to recover the unknown hazard. The
generic estimator is consistent if the model is correctly specified;
alternatively, an oracle inequality can be demonstrated for tree-based models.
To avoid overfitting, boosting employs several regularization devices. One of
them is step-size restriction, but the rationale for this is somewhat
mysterious from the viewpoint of consistency. Our work brings some clarity to
this issue by revealing that step-size restriction is a mechanism for
preventing the curvature of the risk from derailing convergence.
