Scalable Sparse Cox's Regression for Large-Scale Survival Data via Broken Adaptive Ridge

Abstract: This paper develops a new scalable sparse Cox regression tool for sparse high-dimensional massive sample size (sHDMSS) survival data. The method is a local $L_0$-penalized Cox regression via repeatedly performing reweighted $L_2$-penalized Cox regression. We show that the resulting estimator enjoys the best of $L_0$- and $L_2$-penalized Cox regressions while overcoming their limitations. Specifically, the estimator is selection consistent, oracle for parameter estimation, and possesses a grouping property for highly correlated covariates. Simulation results suggest that when the sample size is large, the proposed method with pre-specified tuning parameters has a comparable or better performance than some popular penalized regression methods. More importantly, because the method naturally enables adaptation of efficient algorithms for massive $L_2$-penalized optimization and does not require costly data driven tuning parameter selection, it has a significant computational advantage for sHDMSS data, offering an average of 5-fold speedup over its closest competitor in empirical studies.