# Why Some Models Resist Unlearning: A Linear Stability Perspective

arXiv: https://arxiv.org/abs/2602.02986 · HTML: https://mlsanity.com/paper/2602.02986/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 32 of its 42 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Placed in

- LLM safety and security / [Machine unlearning](https://mlsanity.com/safety/machine-unlearning/)

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

- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Eternal Sunshine of the Spotless Net: Selective Forgetting in Deep Networks](https://mlsanity.com/paper/1911.04933/) (2019)
- [Making AI Forget You: Data Deletion in Machine Learning](https://mlsanity.com/paper/1907.05012/) (2019)
- [Does Learning Require Memorization? A Short Tale about a Long Tail](https://mlsanity.com/paper/1906.05271/) (2019)
- [Remember What You Want to Forget: Algorithms for Machine Unlearning](https://mlsanity.com/paper/2103.03279/) (2021)
- [Amnesiac Machine Learning](https://mlsanity.com/paper/2010.10981/) (2020)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Approximate Data Deletion from Machine Learning Models](https://mlsanity.com/paper/2002.10077/) (2020)
- [Mixed-Privacy Forgetting in Deep Networks](https://mlsanity.com/paper/2012.13431/) (2020)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [Emergent and Predictable Memorization in Large Language Models](https://mlsanity.com/paper/2304.11158/) (2023)
- [The alignment property of SGD noise and how it helps select flat minima: A stability analysis](https://mlsanity.com/paper/2207.02628/) (2022)
- [On Linear Stability of SGD and Input-Smoothness of Neural Networks](https://mlsanity.com/paper/2105.13462/) (2021)
- [Langevin Unlearning: A New Perspective of Noisy Gradient Descent for Machine Unlearning](https://mlsanity.com/paper/2401.10371/) (2024)
- [The Implicit Regularization of Dynamical Stability in Stochastic Gradient Descent](https://mlsanity.com/paper/2305.17490/) (2023)
- [Distribution Density, Tails, and Outliers in Machine Learning: Metrics and Applications](https://mlsanity.com/paper/1910.13427/) (2019)
- [Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalization](https://mlsanity.com/paper/2307.11007/) (2023)
- [Benign Overfitting for Two-layer ReLU Convolutional Neural Networks](https://mlsanity.com/paper/2303.04145/) (2023)
- [Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD](https://mlsanity.com/paper/2412.20553/) (2024)
- [DP-AdamBC: Your DP-Adam Is Actually DP-SGD (Unless You Apply Bias Correction)](https://mlsanity.com/paper/2312.14334/) (2023)
- [SAM operates far from home: eigenvalue regularization as a dynamical phenomenon](https://mlsanity.com/paper/2302.08692/) (2023)
- [Label-Agnostic Forgetting: A Supervision-Free Unlearning in Deep Models](https://mlsanity.com/paper/2404.00506/) (2024)
- [CaMU: Disentangling Causal Effects in Deep Model Unlearning](https://mlsanity.com/paper/2401.17504/) (2024)
- [Understanding Fine-tuning in Approximate Unlearning: A Theoretical Perspective](https://mlsanity.com/paper/2410.03833/) (2024)
- [Skewed Memorization in Large Language Models: Quantification and Decomposition](https://mlsanity.com/paper/2502.01187/) (2025)
- [A Precise Characterization of SGD Stability Using Loss Surface Geometry](https://mlsanity.com/paper/2401.12332/) (2024)

## Abstract (arXiv)

Machine unlearning, the ability to erase the effect of specific training samples without retraining from scratch, is critical for privacy, regulation, and efficiency. However, most progress in unlearning has been empirical, with little theoretical understanding of when and why unlearning works. We tackle this gap by framing unlearning through the lens of asymptotic linear stability to capture the interaction between optimization dynamics and data geometry. The key quantity in our analysis is data coherence which is the cross sample alignment of loss surface directions near the optimum. We decompose coherence along three axes: within the retain set, within the forget set, and between them, and prove tight stability thresholds that separate convergence from divergence. To further link data properties to forgettability, we study a two layer ReLU CNN under a signal plus noise model and show that stronger memorization makes forgetting easier: when the signal to noise ratio (SNR) is lower, cross sample alignment is weaker, reducing coherence and making unlearning easier; conversely, high SNR, highly aligned models resist unlearning. For empirical verification, we show that Hessian tests and CNN heatmaps align closely with the predicted boundary, mapping the stability frontier of gradient based unlearning as a function of batching, mixing, and data/model alignment. Our analysis is grounded in random matrix theory tools and provides the first principled account of the trade offs between memorization, coherence, and unlearning.