# How Hard Can It Be? Hardness-Aware Multi-Objective Unlearning

arXiv: https://arxiv.org/abs/2606.02119 · HTML: https://mlsanity.com/paper/2606.02119/

## 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 35 of its 48 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Placed in

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

## Builds on (most cited first, 30 of 35)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Machine Unlearning](https://mlsanity.com/paper/1912.03817/) (2019)
- [Certified Data Removal from Machine Learning Models](https://mlsanity.com/paper/1911.03030/) (2019)
- [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)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Knowledge Unlearning for Mitigating Privacy Risks in Language Models](https://mlsanity.com/paper/2210.01504/) (2022)
- [Detecting Pretraining Data from Large Language Models](https://mlsanity.com/paper/2310.16789/) (2023)
- [Who's Harry Potter? Approximate Unlearning in LLMs](https://mlsanity.com/paper/2310.02238/) (2023)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Descent-to-Delete: Gradient-Based Methods for Machine Unlearning](https://mlsanity.com/paper/2007.02923/) (2020)
- [Layer by Layer: Uncovering Hidden Representations in Language Models](https://mlsanity.com/paper/2502.02013/) (2025)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Towards Unbounded Machine Unlearning](https://mlsanity.com/paper/2302.09880/) (2023)
- [In-Context Unlearning: Language Models as Few Shot Unlearners](https://mlsanity.com/paper/2310.07579/) (2023)
- [Unlearn What You Want to Forget: Efficient Unlearning for LLMs](https://mlsanity.com/paper/2310.20150/) (2023)
- [Simplicity Prevails: Rethinking Negative Preference Optimization for LLM Unlearning](https://mlsanity.com/paper/2410.07163/) (2024)
- [DeltaGrad: Rapid retraining of machine learning models](https://mlsanity.com/paper/2006.14755/) (2020)
- [Machine Unlearning via Algorithmic Stability](https://mlsanity.com/paper/2102.13179/) (2021)
- [Variational Bayesian Unlearning](https://mlsanity.com/paper/2010.12883/) (2020)
- [Federated Unlearning: How to Efficiently Erase a Client in FL?](https://mlsanity.com/paper/2207.05521/) (2022)
- [What makes unlearning hard and what to do about it](https://mlsanity.com/paper/2406.01257/) (2024)
- [Unlearning Concepts in Diffusion Model via Concept Domain Correction and Concept Preserving Gradient](https://mlsanity.com/paper/2405.15304/) (2024)
- [Backdoor Defense with Machine Unlearning](https://mlsanity.com/paper/2201.09538/) (2022)
- [Rethinking LLM Unlearning Objectives: A Gradient Perspective and Go Beyond](https://mlsanity.com/paper/2502.19301/) (2025)
- [A Comprehensive Survey of Foundation Models in Medicine](https://mlsanity.com/paper/2406.10729/) (2024)
- [Unlearning as multi-task optimization: A normalized gradient difference approach with an adaptive learning rate](https://mlsanity.com/paper/2410.22086/) (2024)
- [To Forget or Not? Towards Practical Knowledge Unlearning for Large Language Models](https://mlsanity.com/paper/2407.01920/) (2024)
- [Markov Chain Monte Carlo-Based Machine Unlearning: Unlearning What Needs to be Forgotten](https://mlsanity.com/paper/2202.13585/) (2022)

## Abstract (arXiv)

Machine unlearning aims to remove the influence of specific forget training data due to privacy, copyright or bias concerns while maintaining the model performance on the remaining retain data. Existing unlearning algorithms, such as optimizing a weighted combination of losses, have tried to achieve these objectives of improving forget quality and maintaining retain utility. However, they do not guarantee that these objectives can be improved by a specified extent for all forget and retain data. In this work, we address this limitation with a novel and theoretically-grounded approach from a constrained optimization perspective. Firstly, we identify that the hardness of reconciling both objectives can be quantified by the similarity between the forget data and the retain data. Next, we derive an unlearning algorithm (HAMU) with the overall goal of guaranteeing a specified improvement in forget quality while minimizing the retain utility cost/degradation by updating the model weights based on our hardness measure. Our hardness measure also informs users when retain utility degradation is unavoidable, i.e., both objectives cannot be improved simultaneously, and stopping should be considered. Our algorithm is applicable to non-convex models and is easily parallelizable, making it readily deployable in real-world scenarios. We empirically demonstrate HAMU's superior performance over baselines on both image and text datasets using large models. Our code is available at https://github.com/aoi3142/HAMU.