# One Sample Fits All: Approximating All Probabilistic Values Simultaneously and Efficiently

arXiv: https://arxiv.org/abs/2410.23808 · HTML: https://mlsanity.com/paper/2410.23808/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Builds on (most cited first, 18 of 18)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Towards Efficient Data Valuation Based on the Shapley Value](https://mlsanity.com/paper/1902.10275/) (2019)
- [Datamodels: Predicting Predictions from Training Data](https://mlsanity.com/paper/2202.00622/) (2022)
- [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)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [Bounding the Estimation Error of Sampling-based Shapley Value Approximation](https://mlsanity.com/paper/1306.4265/) (2013)
- [SHAP-IQ: Unified Approximation of any-order Shapley Interactions](https://mlsanity.com/paper/2303.01179/) (2023)
- [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)
- [SHAQ: Incorporating Shapley Value Theory into Multi-Agent Q-Learning](https://mlsanity.com/paper/2105.15013/) (2021)
- [Understanding Influence Functions and Datamodels via Harmonic Analysis](https://mlsanity.com/paper/2210.01072/) (2022)
- [Provably Accurate Shapley Value Estimation via Leverage Score Sampling](https://mlsanity.com/paper/2410.01917/) (2024)
- [Variance reduced Shapley value estimation for trustworthy data valuation](https://mlsanity.com/paper/2210.16835/) (2022)
- [A Note on "Towards Efficient Data Valuation Based on the Shapley Value''](https://mlsanity.com/paper/2302.11431/) (2023)

## Built on by (most cited first, 9 of 9)

- [Regression-adjusted Monte Carlo Estimators for Shapley Values and Probabilistic Values](https://mlsanity.com/paper/2506.11849/) (2025)
- [Explaining Similarity in Vision-Language Encoders with Weighted Banzhaf Interactions](https://mlsanity.com/paper/2508.05430/) (2025)
- [TreeGrad-Ranker: Feature Ranking via O(L)-Time Gradients for Decision Trees](https://mlsanity.com/paper/2602.11623/) (2026)
- [Faithful Group Shapley Value](https://mlsanity.com/paper/2505.19013/) (2025)
- [Priority-Aware Shapley Value](https://mlsanity.com/paper/2602.09326/) (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)
- [Generalized Priority-Aware Shapley Value](https://mlsanity.com/paper/2605.15018/) (2026)
- [Nonlinear Axiomatic Attribution for Cooperative Games](https://mlsanity.com/paper/2607.09869/) (2026)

## Abstract (arXiv)

The concept of probabilistic values, such as Beta Shapley values and weighted Banzhaf values, has gained recent attention in applications like feature attribution and data valuation. However, exact computation of these values is often exponentially expensive, necessitating approximation techniques. Prior research has shown that the choice of probabilistic values significantly impacts downstream performance, with no universally superior option. Consequently, one may have to approximate multiple candidates and select the best-performing one. Although there have been many efforts to develop efficient estimators, none are intended to approximate all probabilistic values both simultaneously and efficiently. In this work, we embark on the first exploration of achieving this goal. Adhering to the principle of maximum sample reuse, we propose a one-sample-fits-all framework parameterized by a sampling vector to approximate intermediate terms that can be converted to any probabilistic value without amplifying scalars. Leveraging the concept of  (ε, δ) -approximation, we theoretically identify a key formula that effectively determines the convergence rate of our framework. By optimizing the sampling vector using this formula, we obtain i) a one-for-all estimator that achieves the currently best time complexity for all probabilistic values on average, and ii) a faster generic estimator with the sampling vector optimally tuned for each probabilistic value. Particularly, our one-for-all estimator achieves the fastest convergence rate on Beta Shapley values, including the well-known Shapley value, both theoretically and empirically. Finally, we establish a connection between probabilistic values and the least square regression used in (regularized) datamodels, showing that our one-for-all estimator can solve a family of datamodels simultaneously.