Individualized and Global Feature Attributions for Gradient Boosted Trees in the Presence of ℓ2\ell_2 Regularization

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 .Comment: 43 pages, 29 figure

On marginal feature attributions of tree-based models

Due to their power and ease of use, tree-based machine learning models have become very popular. To interpret these models, local feature attributions based on marginal expectations e.g. marginal (interventional) Shapley, Owen or Banzhaf values may be employed. Such feature attribution methods are true to the model and implementation invariant, i.e. dependent only on the input-output function of the model. By taking advantage of the internal structure of tree-based models, we prove that their marginal Shapley values, or more generally marginal feature attributions obtained from a linear game value, are simple (piecewise-constant) functions with respect to a certain finite partition of the input space determined by the trained model. The same is true for feature attributions obtained from the famous TreeSHAP algorithm. Nevertheless, we show that the "path-dependent" TreeSHAP is not implementation invariant by presenting two (statistically similar) decision trees computing the exact same function for which the algorithm yields different rankings of features, whereas the marginal Shapley values coincide. Furthermore, we discuss how the fact that marginal feature attributions are simple functions can potentially be utilized to compute them. An important observation, showcased by experiments with XGBoost, LightGBM and CatBoost libraries, is that only a portion of all features appears in a tree from the ensemble; thus the complexity of computing marginal Shapley (or Owen or Banzhaf) feature attributions may be reduced. In particular, in the case of CatBoost models, the trees are oblivious (symmetric) and the number of features in each of them is no larger than the depth. We exploit the symmetry to derive an explicit formula with improved complexity for marginal Shapley (and Banzhaf and Owen) values which is only in terms of the internal parameters of the CatBoost model.Comment: 48 pages, 7 figure