# Quadrature-TreeSHAP: Depth-Independent TreeSHAP and Shapley Interactions

arXiv: https://arxiv.org/abs/2605.04497 · HTML: https://mlsanity.com/paper/2605.04497/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 57 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 19 of 19)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [Data Banzhaf: A Robust Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2205.15466/) (2022)
- [On the Tractability of SHAP Explanations](https://mlsanity.com/paper/2009.08634/) (2020)
- [SHAP-IQ: Unified Approximation of any-order Shapley Interactions](https://mlsanity.com/paper/2303.01179/) (2023)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [CatBoost: unbiased boosting with categorical features](https://mlsanity.com/paper/1706.09516/) (2017)
- [Beyond TreeSHAP: Efficient Computation of Any-Order Shapley Interactions for Tree Ensembles](https://mlsanity.com/paper/2401.12069/) (2024)
- [Fast TreeSHAP: Accelerating SHAP Value Computation for Trees](https://mlsanity.com/paper/2109.09847/) (2021)
- [GPUTreeShap: Massively Parallel Exact Calculation of SHAP Scores for Tree Ensembles](https://mlsanity.com/paper/2010.13972/) (2020)
- [Linear TreeShap](https://mlsanity.com/paper/2209.08192/) (2022)
- [TsSHAP: Robust model agnostic feature-based explainability for time series forecasting](https://mlsanity.com/paper/2303.12316/) (2023)
- [Fast Estimation of Partial Dependence Functions using Trees](https://mlsanity.com/paper/2410.13448/) (2024)
- [HyperSHAP: Shapley Values and Interactions for Explaining Hyperparameter Optimization](https://mlsanity.com/paper/2502.01276/) (2025)
- [TreeGrad-Ranker: Feature Ranking via O(L)-Time Gradients for Decision Trees](https://mlsanity.com/paper/2602.11623/) (2026)
- [On marginal feature attributions of tree-based models](https://mlsanity.com/paper/2302.08434/) (2023)
- [From Decision Trees to Boolean Logic: A Fast and Unified SHAP Algorithm](https://mlsanity.com/paper/2511.09376/) (2025)

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

- [QuadraSHAP: ε-Exact Shapley Values for Product Games in Logarithmic Parallel Time](https://mlsanity.com/paper/2605.05870/) (2026)
- [qshap: Fast Shapley Decomposition of R2 for Gradient-Boosted Trees](https://mlsanity.com/paper/2608.24104/) (2026)

## Abstract (arXiv)

Shapley values are a standard tool for explaining predictions of tree ensembles, with Path-Dependent SHAP being the most widely used variant. Despite substantial progress, existing methods still exhibit trade-offs between depth-dependent runtime, numerical stability, and support for higher-order interactions. To address these challenges, we introduce Quadrature-TreeSHAP, a quadrature-based reformulation of Path-Dependent TreeSHAP that is numerically stable, naturally extends to any-order Shapley interaction values and is practically insensitive to tree depth. Our implementation supports both CPU and GPU and is integrated into XGBoost. Our method is based on a weighted-Banzhaf interaction polynomial, which expresses Banzhaf interaction values as expectations under a feature participation probability p. Shapley values and any-order interaction values are then recovered by integrating these polynomials over p from 0 to 1. We evaluate these integrals using Gauss-Legendre quadrature, and show that, in practice, only 8 fixed quadrature points are sufficient to reach machine precision. In fact, Quadrature-TreeSHAP with 8 fixed points achieves greater numerical stability than TreeSHAP. This fixed-point formulation removes depth dependence from the inner computation and enables efficient SIMD execution. We confirm these advantages empirically. On 12 XGBoost benchmarks, Quadrature-TreeSHAP computes Shapley values 1.06x-10.59x faster than TreeSHAP on CPU and 1.84x-6.95x faster than GPUTreeSHAP on GPU. Shapley pairwise interactions are 3.80x-58.11x faster on CPU, with higher-order interactions achieving speedups of up to 1200x compared to TreeSHAP-IQ.