A Gradient Boosting Algorithm for Survival Analysis via Direct Optimization of Concordance Index

Algorithm 1

(Stochastic) gradient boosting machine for concordance index learning (GBMCI).

Initialize {𝑓₀(𝑥_𝑖)}^𝑛_𝑖=1 with the prediction of Cox's PH model

Set shrinkage 𝜈, and subsampling size 𝑛_𝑠≤𝑛

For  𝑚=1∶𝑀  do

 Compute {𝑔_𝑖𝑚}^𝑛_𝑖=1 by (9)

 Randomly select a subset {𝑥_𝑖,𝑡_𝑖,𝛿_𝑖}^𝑛𝑠_𝑖=1 from the whole dataset

 Fit the weak learner 𝑓(𝑥;𝜏_𝑚), for example, a regression tree, upon {𝑥_𝑖,𝑔_𝑖𝑚}^𝑛𝑠_𝑖=1

 Compute 𝜌_𝑚 by (10) using line-search

 Update {𝐹_𝑚(𝑥_𝑖)}^𝑛_𝑖=1 by 𝐹_𝑚(𝑥)=𝐹_𝑚−1(𝑥)+𝜈𝜌_𝑚𝑓(𝑥;𝜏_𝑚)

end for