  We consider the finite sample properties of the regularized high-dimensional
Cox regression via lasso. Existing literature focuses on linear models or
generalized linear models with Lipschitz loss functions, where the empirical
risk functions are the summations of independent and identically distributed
(iid) losses. The summands in the negative log partial likelihood function for
censored survival data, however, are neither iid nor Lipschitz. We first
approximate the negative log partial likelihood function by a sum of iid
non-Lipschitz terms, then derive the non-asymptotic oracle inequalities for the
lasso penalized Cox regression using pointwise arguments to tackle the
difficulty caused by the lack of iid and Lipschitz property.
