  We consider a general high-dimensional additive hazard model in a
non-asymptotic setting, including regression for censored-data. In this
context, we consider a Lasso estimator with a fully data-driven $\ell_1$
penalization, which is tuned for the estimation problem at hand. We prove sharp
oracle inequalities for this estimator. Our analysis involves a new
"data-driven" Bernstein's inequality, that is of independent interest, where
the predictable variation is replaced by the optional variation.
