# Influence Diagnostics in High-dimensional M-estimation: Precise Asymptotics

arXiv: https://arxiv.org/abs/2607.09250 · HTML: https://mlsanity.com/paper/2607.09250/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 61 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 22 of 22)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Influence Functions in Deep Learning Are Fragile](https://mlsanity.com/paper/2006.14651/) (2020)
- [High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification](https://mlsanity.com/paper/1507.03003/) (2015)
- [If Influence Functions are the Answer, Then What is the Question?](https://mlsanity.com/paper/2209.05364/) (2022)
- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [A Statistical Perspective on Algorithmic Leveraging](https://mlsanity.com/paper/1306.5362/) (2013)
- [Most Influential Subset Selection: Challenges, Promises, and Beyond](https://mlsanity.com/paper/2409.18153/) (2024)
- [Optimal Subsampling for Large Sample Logistic Regression](https://mlsanity.com/paper/1702.01166/) (2017)
- [An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference?](https://mlsanity.com/paper/2011.14999/) (2020)
- [Towards a statistical theory of data selection under weak supervision](https://mlsanity.com/paper/2309.14563/) (2023)
- [Improving the Scaling Laws of Synthetic Data with Deliberate Practice](https://mlsanity.com/paper/2502.15588/) (2025)
- [High-dimensional robust regression under heavy-tailed data: Asymptotics and Universality](https://mlsanity.com/paper/2309.16476/) (2023)
- [Why Less is More (Sometimes): A Theory of Data Curation](https://mlsanity.com/paper/2511.03492/) (2025)
- [Large deviations for the perceptron model and consequences for active learning](https://mlsanity.com/paper/1912.03927/) (2019)
- [Observable adjustments in single-index models for regularized M-estimators](https://mlsanity.com/paper/2204.06990/) (2022)
- [Local minima of the empirical risk in high dimension: General theorems and convex examples](https://mlsanity.com/paper/2502.01953/) (2025)
- [High-Dimensional Statistics: Reflections on Progress and Open Problems](https://mlsanity.com/paper/2605.05076/) (2026)
- [Testing Most Influential Sets](https://mlsanity.com/paper/2510.20372/) (2025)
- [Asymptotic Theory of Iterated Empirical Risk Minimization, with Applications to Active Learning](https://mlsanity.com/paper/2601.23031/) (2026)
- [Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities](https://mlsanity.com/paper/2102.08020/) (2021)
- [Statistical and Computational Guarantees for Influence Diagnostics](https://mlsanity.com/paper/2212.04014/) (2022)

## Abstract (arXiv)

The impact of a given training point on a statistical model can be measured through its leave-one-out influence on the model parameters, which quantifies how its removal from the training set affects the learned weights. For convex M-estimation under Gaussian design, in the high-dimensional limit nasymp d, we show that the empirical distribution of influences across training points concentrates around a deterministic measure which we sharply characterize. This characterization suggests that influential samples tend to lie on average close to the decision boundary, making contact with a standard data selection heuristic in active learning.