# Rethinking Influence Functions of Neural Networks in the Over-parameterized Regime

arXiv: https://arxiv.org/abs/2112.08297 · HTML: https://mlsanity.com/paper/2112.08297/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 32 references that are in the map.
- First posted on arXiv 15 Dec 2021.

## Builds on (most cited first, 23 of 23)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Influence Functions in Deep Learning Are Fragile](https://mlsanity.com/paper/2006.14651/) (2020)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [How Does Mixup Help With Robustness and Generalization?](https://mlsanity.com/paper/2010.04819/) (2020)
- [Graph Neural Tangent Kernel: Fusing Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/1905.13192/) (2019)
- [Gradient Descent with Early Stopping is Provably Robust to Label Noise for Overparameterized Neural Networks](https://mlsanity.com/paper/1903.11680/) (2019)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [On the Accuracy of Influence Functions for Measuring Group Effects](https://mlsanity.com/paper/1905.13289/) (2019)
- [Enhanced Convolutional Neural Tangent Kernels](https://mlsanity.com/paper/1911.00809/) (2019)
- [Global Convergence of Deep Networks with One Wide Layer Followed by Pyramidal Topology](https://mlsanity.com/paper/2002.07867/) (2020)
- [Repairing without Retraining: Avoiding Disparate Impact with Counterfactual Distributions](https://mlsanity.com/paper/1901.10501/) (2019)
- [HYDRA: Hypergradient Data Relevance Analysis for Interpreting Deep Neural Networks](https://mlsanity.com/paper/2102.02515/) (2021)
- [Diverse Neural Network Learns True Target Functions](https://mlsanity.com/paper/1611.03131/) (2016)
- [Distillation ≈ Early Stopping? Harvesting Dark Knowledge Utilizing Anisotropic Information Retrieval For Overparameterized Neural Network](https://mlsanity.com/paper/1910.01255/) (2019)
- [Data Dropout: Optimizing Training Data for Convolutional Neural Networks](https://mlsanity.com/paper/1809.00193/) (2018)
- [Understanding Instance-based Interpretability of Variational Auto-Encoders](https://mlsanity.com/paper/2105.14203/) (2021)

## Built on by (most cited first, 26 of 26)

- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [In-Context Unlearning: Language Models as Few Shot Unlearners](https://mlsanity.com/paper/2310.07579/) (2023)
- [Training Data Influence Analysis and Estimation: A Survey](https://mlsanity.com/paper/2212.04612/) (2022)
- [Machine Unlearning Fails to Remove Data Poisoning Attacks](https://mlsanity.com/paper/2406.17216/) (2024)
- [Data Valuation Without Training of a Model](https://mlsanity.com/paper/2301.00930/) (2023)
- [Identifying a Training-Set Attack's Target Using Renormalized Influence Estimation](https://mlsanity.com/paper/2201.10055/) (2022)
- [Reconstruction Attacks on Machine Unlearning: Simple Models are Vulnerable](https://mlsanity.com/paper/2405.20272/) (2024)
- [Understanding Influence Functions and Datamodels via Harmonic Analysis](https://mlsanity.com/paper/2210.01072/) (2022)
- [Regularizing Second-Order Influences for Continual Learning](https://mlsanity.com/paper/2304.10177/) (2023)
- [On the Trade-Off between Actionable Explanations and the Right to be Forgotten](https://mlsanity.com/paper/2208.14137/) (2022)
- [A Bayesian Approach To Analysing Training Data Attribution In Deep Learning](https://mlsanity.com/paper/2305.19765/) (2023)
- [Generalization Through The Lens Of Leave-One-Out Error](https://mlsanity.com/paper/2203.03443/) (2022)
- [Sample based Explanations via Generalized Representers](https://mlsanity.com/paper/2310.18526/) (2023)
- [Final-Model-Only Data Attribution with a Unifying View of Gradient-Based Methods](https://mlsanity.com/paper/2412.03906/) (2024)
- [Taming Hyperparameter Sensitivity in Data Attribution: Practical Selection Without Costly Retraining](https://mlsanity.com/paper/2505.24261/) (2025)
- [The Unseen Threat: Residual Knowledge in Machine Unlearning under Perturbed Samples](https://mlsanity.com/paper/2601.22359/) (2026)
- [UtilGen: Utility-Centric Generative Data Augmentation with Dual-Level Task Adaptation](https://mlsanity.com/paper/2510.24262/) (2025)
- [OTAD: An Optimal Transport-Induced Robust Model for Agnostic Adversarial Attack](https://mlsanity.com/paper/2408.00329/) (2024)
- [PINNfluence: Interpreting PINNs through Influence Functions](https://mlsanity.com/paper/2409.08958/) (2024)
- [Where Did Your Model Learn That? Label-free Influence for Self-supervised Learning](https://mlsanity.com/paper/2412.17170/) (2024)
- [Flexible Concept Bottleneck Model](https://mlsanity.com/paper/2511.06678/) (2025)
- [On the Trade-offs between Adversarial Robustness and Actionable Explanations](https://mlsanity.com/paper/2309.16452/) (2023)
- [Interpretable Feature Interaction via Statistical Self-supervised Learning on Tabular Data](https://mlsanity.com/paper/2503.18048/) (2025)
- [A Comparative Analysis of Influence Signals for Data Debugging](https://mlsanity.com/paper/2506.11584/) (2025)
- [f-INE: A Hypothesis Testing Framework for Estimating Influence under Training Randomness](https://mlsanity.com/paper/2510.10510/) (2025)
- [Linearized Attention Cannot Enter the Kernel Regime at Any Practical Width](https://mlsanity.com/paper/2603.13085/) (2026)

## Abstract (arXiv)

Understanding the black-box prediction for neural networks is challenging. To achieve this, early studies have designed influence function (IF) to measure the effect of removing a single training point on neural networks. However, the classic implicit Hessian-vector product (IHVP) method for calculating IF is fragile, and theoretical analysis of IF in the context of neural networks is still lacking. To this end, we utilize the neural tangent kernel (NTK) theory to calculate IF for the neural network trained with regularized mean-square loss, and prove that the approximation error can be arbitrarily small when the width is sufficiently large for two-layer ReLU networks. We analyze the error bound for the classic IHVP method in the over-parameterized regime to understand when and why it fails or not. In detail, our theoretical analysis reveals that (1) the accuracy of IHVP depends on the regularization term, and is pretty low under weak regularization; (2) the accuracy of IHVP has a significant correlation with the probability density of corresponding training points. We further borrow the theory from NTK to understand the IFs better, including quantifying the complexity for influential samples and depicting the variation of IFs during the training dynamics. Numerical experiments on real-world data confirm our theoretical results and demonstrate our findings.