# SAUL: Sharpness-Aware Augmented-Lagrangian Unlearning

arXiv: https://arxiv.org/abs/2608.16249 · HTML: https://mlsanity.com/paper/2608.16249/

## 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 16 of its 24 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Placed in

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

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

- [Extracting Training Data from Large Language Models](https://mlsanity.com/paper/2012.07805/) (2020)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [TOFU: A Task of Fictitious Unlearning for LLMs](https://mlsanity.com/paper/2401.06121/) (2024)
- [Negative Preference Optimization: From Catastrophic Collapse to Effective Unlearning](https://mlsanity.com/paper/2404.05868/) (2024)
- [Knowledge Unlearning for Mitigating Privacy Risks in Language Models](https://mlsanity.com/paper/2210.01504/) (2022)
- [Who's Harry Potter? Approximate Unlearning in LLMs](https://mlsanity.com/paper/2310.02238/) (2023)
- [MUSE: Machine Unlearning Six-Way Evaluation for Language Models](https://mlsanity.com/paper/2407.06460/) (2024)
- [Sharpness-Aware Minimization Improves Language Model Generalization](https://mlsanity.com/paper/2110.08529/) (2021)
- [Towards LLM Unlearning Resilient to Relearning Attacks: A Sharpness-Aware Minimization Perspective and Beyond](https://mlsanity.com/paper/2502.05374/) (2025)
- [LLM Unlearning via Loss Adjustment with Only Forget Data](https://mlsanity.com/paper/2410.11143/) (2024)
- [Unlearning as multi-task optimization: A normalized gradient difference approach with an adaptive learning rate](https://mlsanity.com/paper/2410.22086/) (2024)
- [BLUR: A Bi-Level Optimization Approach for LLM Unlearning](https://mlsanity.com/paper/2506.08164/) (2025)
- [Sharpness-Aware Machine Unlearning](https://mlsanity.com/paper/2506.13715/) (2025)
- [DualOptim: Enhancing Efficacy and Stability in Machine Unlearning with Dual Optimizers](https://mlsanity.com/paper/2504.15827/) (2025)
- [OFMU: Optimization-Driven Framework for Machine Unlearning](https://mlsanity.com/paper/2509.22483/) (2025)
- [Sharpness-Aware Parameter Selection for Machine Unlearning](https://mlsanity.com/paper/2504.06398/) (2025)

## Abstract (arXiv)

Machine unlearning in Large Language Models (LLMs) faces a critical trade-off between erasing target knowledge and preserving general utility. We propose SAUL (Sharpness-Aware Augmented-Lagrangian Unlearning), which formulates unlearning as a constrained minimization problem following the principle of "forget enough, but no more than necessary." At its core, SAUL formulates forgetting as an explicit constraint with a prescribed satisfaction criterion, whereas prior unlearning methods typically specify the desired level of forgetting implicitly through optimization objectives. An augmented Lagrangian controller adaptively adjusts forget-side pressure according to constraint violation and can eventually deactivate the forget-side update as the prescribed criterion remains satisfied. Sharpness-aware updates on both retain and forget objectives, together with a dual-optimizer design that maintains role-separated states, further stabilize the resulting unlearning dynamics. We evaluate SAUL on the TOFU, WMDP, and MUSE benchmarks, demonstrating favorable forgetting-utility trade-offs over representative sharpness- and perturbation-based baselines under benchmark-specific forgetting criteria. Beyond the complete SAUL framework, we further show on TOFU that applying the augmented-Lagrangian controller as a drop-in modifier to representative baselines improves their post-forgetting utility, demonstrating the practical value of explicit forgetting control.