# The Memory Perturbation Equation: Understanding Model's Sensitivity to Data

arXiv: https://arxiv.org/abs/2310.19273 · HTML: https://mlsanity.com/paper/2310.19273/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 57 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 30 of 32)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [An Empirical Study of Example Forgetting during Deep Neural Network Learning](https://mlsanity.com/paper/1812.05159/) (2018)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Deep Learning on a Data Diet: Finding Important Examples Early in Training](https://mlsanity.com/paper/2107.07075/) (2021)
- [What Neural Networks Memorize and Why: Discovering the Long Tail via Influence Estimation](https://mlsanity.com/paper/2008.03703/) (2020)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Not All Samples Are Created Equal: Deep Learning with Importance Sampling](https://mlsanity.com/paper/1803.00942/) (2018)
- [Influence Functions in Deep Learning Are Fragile](https://mlsanity.com/paper/2006.14651/) (2020)
- [Sensitivity and Generalization in Neural Networks: an Empirical Study](https://mlsanity.com/paper/1802.08760/) (2018)
- [On the Accuracy of Influence Functions for Measuring Group Effects](https://mlsanity.com/paper/1905.13289/) (2019)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning](https://mlsanity.com/paper/2104.04975/) (2021)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Estimating Example Difficulty Using Variance of Gradients](https://mlsanity.com/paper/2008.11600/) (2020)
- [To Repeat or Not To Repeat: Insights from Scaling LLM under Token-Crisis](https://mlsanity.com/paper/2305.13230/) (2023)
- [Approximate Inference Turns Deep Networks into Gaussian Processes](https://mlsanity.com/paper/1906.01930/) (2019)
- [Data Cleansing for Models Trained with SGD](https://mlsanity.com/paper/1906.08473/) (2019)
- [Conjugate-Computation Variational Inference : Converting Variational Inference in Non-Conjugate Models to Inferences in Conjugate Models](https://mlsanity.com/paper/1703.04265/) (2017)
- [Filter Response Normalization Layer: Eliminating Batch Dependence in the Training of Deep Neural Networks](https://mlsanity.com/paper/1911.09737/) (2019)
- [One Epoch Is All You Need](https://mlsanity.com/paper/1906.06669/) (2019)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models](https://mlsanity.com/paper/1803.09151/) (2018)
- [DeepOBS: A Deep Learning Optimizer Benchmark Suite](https://mlsanity.com/paper/1903.05499/) (2019)
- [Handling the Positive-Definite Constraint in the Bayesian Learning Rule](https://mlsanity.com/paper/2002.10060/) (2020)
- [Repairing Neural Networks by Leaving the Right Past Behind](https://mlsanity.com/paper/2207.04806/) (2022)
- [ASDL: A Unified Interface for Gradient Preconditioning in PyTorch](https://mlsanity.com/paper/2305.04684/) (2023)
- [Estimating informativeness of samples with Smooth Unique Information](https://mlsanity.com/paper/2101.06640/) (2021)

## Built on by (most cited first, 16 of 16)

- [Variational Learning is Effective for Large Deep Networks](https://mlsanity.com/paper/2402.17641/) (2024)
- [Training Data Attribution via Approximate Unrolled Differentiation](https://mlsanity.com/paper/2405.12186/) (2024)
- [Generative Uncertainty in Diffusion Models](https://mlsanity.com/paper/2502.20946/) (2025)
- [Trade-Offs of Diagonal Fisher Information Matrix Estimators](https://mlsanity.com/paper/2402.05379/) (2024)
- [Explainable Natural Language Processing for Corporate Sustainability Analysis](https://mlsanity.com/paper/2407.17487/) (2024)
- [Knowledge Adaptation as Posterior Correction](https://mlsanity.com/paper/2506.14262/) (2025)
- [Revisiting the robustness of post-hoc interpretability methods](https://mlsanity.com/paper/2407.19683/) (2024)
- [Variational Low-Rank Adaptation Using IVON](https://mlsanity.com/paper/2411.04421/) (2024)
- [Variational Routing: A Scalable Bayesian Framework for Calibrated Mixture-of-Experts Transformers](https://mlsanity.com/paper/2603.09453/) (2026)
- [SOAP-Bubbles: Structured Weight Uncertainty for Neural Networks](https://mlsanity.com/paper/2606.23357/) (2026)
- [DICE: Data Influence Cascade in Decentralized Learning](https://mlsanity.com/paper/2507.06931/) (2025)
- [Approximating Full Conformal Prediction for Neural Network Regression with Gauss-Newton Influence](https://mlsanity.com/paper/2507.20272/) (2025)
- [Set-Valued Sensitivity Analysis of Deep Neural Networks](https://mlsanity.com/paper/2412.11057/) (2024)
- [Variational Learning Induces Adaptive Label Smoothing](https://mlsanity.com/paper/2502.07273/) (2025)
- [Joint Model and Data Sparsification via the Marginal Likelihood](https://mlsanity.com/paper/2605.29908/) (2026)
- [Quantifying the Agreement Between Data-Influence and Data-Similarity to Understand LLM Behavior](https://mlsanity.com/paper/2606.23591/) (2026)

## Abstract (arXiv)

Understanding model's sensitivity to its training data is crucial but can also be challenging and costly, especially during training. To simplify such issues, we present the Memory-Perturbation Equation (MPE) which relates model's sensitivity to perturbation in its training data. Derived using Bayesian principles, the MPE unifies existing sensitivity measures, generalizes them to a wide-variety of models and algorithms, and unravels useful properties regarding sensitivities. Our empirical results show that sensitivity estimates obtained during training can be used to faithfully predict generalization on unseen test data. The proposed equation is expected to be useful for future research on robust and adaptive learning.