# Testing Most Influential Sets

arXiv: https://arxiv.org/abs/2510.20372 · HTML: https://mlsanity.com/paper/2510.20372/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 23 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [On the Accuracy of Influence Functions for Measuring Group Effects](https://mlsanity.com/paper/1905.13289/) (2019)
- [Why Is My Classifier Discriminatory?](https://mlsanity.com/paper/1805.12002/) (2018)
- [On Second-Order Group Influence Functions for Black-Box Predictions](https://mlsanity.com/paper/1911.00418/) (2019)
- [Leave-one-out Unfairness](https://mlsanity.com/paper/2107.10171/) (2021)
- [Most Influential Subset Selection: Challenges, Promises, and Beyond](https://mlsanity.com/paper/2409.18153/) (2024)
- [Theoretical and Practical Perspectives on what Influence Functions Do](https://mlsanity.com/paper/2305.16971/) (2023)
- [An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference?](https://mlsanity.com/paper/2011.14999/) (2020)
- [Fair Infinitesimal Jackknife: Mitigating the Influence of Biased Training Data Points Without Refitting](https://mlsanity.com/paper/2212.06803/) (2022)
- [Hardness and Algorithms for Robust and Sparse Optimization](https://mlsanity.com/paper/2206.14354/) (2022)
- [Approximations to worst-case data dropping: unmasking failure modes](https://mlsanity.com/paper/2408.09008/) (2024)
- [Provably Auditing Ordinary Least Squares in Low Dimensions](https://mlsanity.com/paper/2205.14284/) (2022)
- [Towards Practical Robustness Auditing for Linear Regression](https://mlsanity.com/paper/2307.16315/) (2023)
- [Robustness Auditing for Linear Regression: To Singularity and Beyond](https://mlsanity.com/paper/2410.07916/) (2024)

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

- [Finding Most Influential Sets](https://mlsanity.com/paper/2606.05919/) (2026)
- [Influence Diagnostics in High-dimensional M-estimation: Precise Asymptotics](https://mlsanity.com/paper/2607.09250/) (2026)

## Abstract (arXiv)

Small influential data subsets can dramatically impact model conclusions, with a few data points overturning key findings. While recent work identifies these most influential sets, there is no formal way to tell when maximum influence is excessive rather than expected under natural random sampling variation. We address this gap by developing a principled framework for most influential sets. Focusing on linear least-squares, we derive a convenient exact influence formula and identify the extreme value distributions of maximal influence - the heavy-tailed Fr'echet for constant-size sets and heavy-tailed data, and the well-behaved Gumbel for growing sets or light tails. This allows us to conduct rigorous hypothesis tests for excessive influence. We demonstrate through applications across economics, biology, and machine learning benchmarks, resolving contested findings and replacing ad-hoc heuristics with rigorous inference.