Non-asymptotic Oracle Inequalities for the High-Dimensional Cox Regression via Lasso
This work addresses a methodological gap for researchers in survival analysis and high-dimensional statistics, though it is incremental as it extends existing lasso theory to a specific non-standard setting.
The paper tackles the challenge of deriving finite-sample guarantees for high-dimensional Cox regression with lasso regularization in censored survival data, where the negative log partial likelihood is non-iid and non-Lipschitz, by approximating it with iid terms and establishing non-asymptotic oracle inequalities.
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.