# Regression-adjusted Monte Carlo Estimators for Shapley Values and Probabilistic Values

arXiv: https://arxiv.org/abs/2506.11849 · HTML: https://mlsanity.com/paper/2506.11849/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 41 references that are in the map.
- First posted on arXiv 13 Jun 2025.

## Builds on (most cited first, 17 of 17)

- [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)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [Feature relevance quantification in explainable AI: A causal problem](https://mlsanity.com/paper/1910.13413/) (2019)
- [Beta Shapley: a Unified and Noise-reduced Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2110.14049/) (2021)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [Data Banzhaf: A Robust Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2205.15466/) (2022)
- [Data Shapley in One Training Run](https://mlsanity.com/paper/2406.11011/) (2024)
- [Measuring the Effect of Training Data on Deep Learning Predictions via Randomized Experiments](https://mlsanity.com/paper/2206.10013/) (2022)
- [WeightedSHAP: analyzing and improving Shapley based feature attributions](https://mlsanity.com/paper/2209.13429/) (2022)
- [Approximating the Shapley Value without Marginal Contributions](https://mlsanity.com/paper/2302.00736/) (2023)
- [Provably Accurate Shapley Value Estimation via Leverage Score Sampling](https://mlsanity.com/paper/2410.01917/) (2024)
- [ProxySPEX: Inference-Efficient Interpretability via Sparse Feature Interactions in LLMs](https://mlsanity.com/paper/2505.17495/) (2025)
- [One Sample Fits All: Approximating All Probabilistic Values Simultaneously and Efficiently](https://mlsanity.com/paper/2410.23808/) (2024)
- [A Unified Framework for Provably Efficient Algorithms to Estimate Shapley Values](https://mlsanity.com/paper/2506.05216/) (2025)
- [Kernel Banzhaf: A Fast and Robust Estimator for Banzhaf Values](https://mlsanity.com/paper/2410.08336/) (2024)

## Built on by (most cited first, 19 of 19)

- [PolySHAP: Extending KernelSHAP with Interaction-Informed Polynomial Regression](https://mlsanity.com/paper/2601.18608/) (2026)
- [An Odd Estimator for Shapley Values](https://mlsanity.com/paper/2602.01399/) (2026)
- [Exactly Computing do-Shapley Values](https://mlsanity.com/paper/2602.07203/) (2026)
- [Tractable Shapley Values and Interactions via Tensor Networks](https://mlsanity.com/paper/2510.22138/) (2025)
- [Priority-Aware Shapley Value](https://mlsanity.com/paper/2602.09326/) (2026)
- [Functional Decomposition and Shapley Interactions for Interpreting Survival Models](https://mlsanity.com/paper/2602.16505/) (2026)
- [Adalina: Adaptive Linear Approximation for the Shapley Value and Beyond](https://mlsanity.com/paper/2604.08438/) (2026)
- [CUBE: Contrastive Understanding by Balanced Experiments](https://mlsanity.com/paper/2509.10825/) (2025)
- [Geometric Data Valuation via Leverage Scores](https://mlsanity.com/paper/2511.02100/) (2025)
- [SurrogateSHAP: Training-Free Contributor Attribution for Text-to-Image (T2I) Models](https://mlsanity.com/paper/2601.22276/) (2026)
- [ConfoundingSHAP: Quantifying confounding strength in causal inference](https://mlsanity.com/paper/2605.10533/) (2026)
- [Generalized Priority-Aware Shapley Value](https://mlsanity.com/paper/2605.15018/) (2026)
- [Verified SHAP: Provable Bounds for Exact Shapley Values of Neural Networks](https://mlsanity.com/paper/2605.24084/) (2026)
- [TN-SHAP-G: Graph-Structured Tensor Network Surrogates for Shapley Values and Interactions](https://mlsanity.com/paper/2606.01540/) (2026)
- [ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation](https://mlsanity.com/paper/2606.02247/) (2026)
- [OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators](https://mlsanity.com/paper/2606.28065/) (2026)
- [ShapKO: Shapley-Adaptive Modality Knockout for Robust Multimodal Learning](https://mlsanity.com/paper/2607.09884/) (2026)
- [RelShap: Relationally Consistent Shapley Explanations](https://mlsanity.com/paper/2608.11508/) (2026)
- [Building Socio-Affective Artificial Intelligence for Interactive Multi-Agent Simulations](https://mlsanity.com/paper/2609.26927/) (2026)

## Abstract (arXiv)

With origins in game theory, probabilistic values like Shapley values, Banzhaf values, and semi-values have emerged as a central tool in explainable AI. They are used for feature attribution, data attribution, data valuation, and more. Since exact computation can require exponential time, research has focused on efficient approximation methods using two techniques: Monte Carlo sampling and linear regression formulations. In this work, we present a new way of combining both of these techniques. Our approach is more flexible than prior algorithms, allowing for linear regression to be replaced with any function family whose probabilistic values can be computed efficiently. This allows us to harness the accuracy of tree-based models like XGBoost; with cross-fitting, the method also produces unbiased estimates. From experiments across eight datasets, we find that our methods give state-of-the-art performance for estimating probabilistic values. For Shapley values, the error of our methods can be 11× lower than Permutation SHAP (the most popular Monte Carlo method), 6.5× lower than Kernel SHAP (the most popular linear regression method), and 3.4× lower than Leverage SHAP (the prior state-of-the-art Shapley value estimator). For more general probabilistic values, we can obtain error 390× lower than the best estimator from prior work.