AXIL: Exact Instance Attribution for Gradient Boosting

We derive an exact, prediction-specific instance-attribution method for fitted gradient boosting machines (GBMs) trained with squared-error loss, with the learned tree structure held fixed. Each prediction can be written as a weighted sum of training targets, with coefficients determined only by the fitted tree structure and learning rate. These coefficients are exact instance attributions, or AXIL weights. Our main algorithmic contribution is a matrix-free backward operator that computes one AXIL attribution vector in O(TN) time, or S vectors in O(TNS), without materialising the full N x N matrix. This extends to out-of-sample predictions and makes exact instance attribution practical for large datasets. AXIL yields exact fixed-structure sensitivity by construction in target-perturbation tests, where competing GBM-specific attribution methods (BoostIn, TREX, and LeafInfluence) generally fail. In retraining-based faithfulness tests on 20 regression datasets, AXIL achieves the highest faithfulness score on 14 datasets and statistically ties for the best on 4 others, while also running substantially faster than the competing methods. We also show that the AXIL weight matrix is the globally constant special case of a target-response Jacobian that provides first-order instance attribution for any differentiable learner via implicit differentiation, placing the exact decomposition inside a broader framework.