# AXIL: Exact Instance Attribution for Gradient Boosting

arXiv: https://arxiv.org/abs/2301.01864 · HTML: https://mlsanity.com/paper/2301.01864/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 29 references that are in the map.
- First posted on arXiv 5 Jan 2023.

## Builds on (most cited first, 14 of 14)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Deep Neural Networks and Tabular Data: A Survey](https://mlsanity.com/paper/2110.01889/) (2021)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Why do tree-based models still outperform deep learning on tabular data?](https://mlsanity.com/paper/2207.08815/) (2022)
- [Datamodels: Predicting Predictions from Training Data](https://mlsanity.com/paper/2202.00622/) (2022)
- [Representer Point Selection for Explaining Deep Neural Networks](https://mlsanity.com/paper/1811.09720/) (2018)
- [Adapting and Evaluating Influence-Estimation Methods for Gradient-Boosted Decision Trees](https://mlsanity.com/paper/2205.00359/) (2022)
- [Finding Influential Training Samples for Gradient Boosted Decision Trees](https://mlsanity.com/paper/1802.06640/) (2018)

## Built on by (most cited first, 2 of 2)

- [Enhanced Local Explainability and Trust Scores with Random Forest Proximities](https://mlsanity.com/paper/2310.12428/) (2023)
- [LGB+: A Macroeconomic Forecasting Road Test](https://mlsanity.com/paper/2605.09740/) (2026)

## Abstract (arXiv)

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.