  While $\ell_2$ regularization is widely used in training gradient boosted
trees, popular individualized feature attribution methods for trees such as
Saabas and TreeSHAP overlook the training procedure. We propose Prediction
Decomposition Attribution (PreDecomp), a novel individualized feature
attribution for gradient boosted trees when they are trained with $\ell_2$
regularization. Theoretical analysis shows that the inner product between
PreDecomp and labels on in-sample data is essentially the total gain of a tree,
and that it can faithfully recover additive models in the population case when
features are independent. Inspired by the connection between PreDecomp and
total gain, we also propose TreeInner, a family of debiased global feature
attributions defined in terms of the inner product between any individualized
feature attribution and labels on out-sample data for each tree. Numerical
experiments on a simulated dataset and a genomic ChIP dataset show that
TreeInner has state-of-the-art feature selection performance. Code reproducing
experiments is available at https://github.com/nalzok/TreeInner .
