  This paper develops two orthogonal contributions to scalable sparse
regression for competing risks time-to-event data. First, we study and
accelerate the broken adaptive ridge method (BAR), an $\ell_0$-based
iteratively reweighted $\ell_2$-penalization algorithm that achieves sparsity
in its limit, in the context of the Fine-Gray (1999) proportional
subdistributional hazards (PSH) model. In particular, we derive a new algorithm
for BAR regression, named cycBAR, that performs cyclic update of each
coordinate using an explicit thresholding formula. The new cycBAR algorithm
effectively avoids fitting multiple reweighted $\ell_2$-penalizations and thus
yields impressive speedups over the original BAR algorithm. Second, we address
a pivotal computational issue related to fitting the PSH model. Specifically,
the computation costs of the log-pseudo likelihood and its derivatives for PSH
model grow at the rate of $O(n^2)$ with the sample size $n$ in current
implementations. We propose a novel forward-backward scan algorithm that
reduces the computation costs to $O(n)$. The proposed method applies to both
unpenalized and penalized estimation for the PSH model and has exhibited
drastic speedups over current implementations. Finally, combining the two
algorithms can yields $>1,000$ fold speedups over the original BAR algorithm.
Illustrations of the impressive scalability of our proposed algorithm for large
competing risks data are given using both simulations and a United States Renal
Data System data.
