# BLUR: A Bi-Level Optimization Approach for LLM Unlearning

arXiv: https://arxiv.org/abs/2506.08164 · HTML: https://mlsanity.com/paper/2506.08164/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 44 of its 61 references that are in the map.
- First posted on arXiv 9 Jun 2025.

## Placed in

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

## Builds on (most cited first, 30 of 44)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Erasing Concepts from Diffusion Models](https://mlsanity.com/paper/2303.07345/) (2023)
- [TOFU: A Task of Fictitious Unlearning for LLMs](https://mlsanity.com/paper/2401.06121/) (2024)
- [The WMDP Benchmark: Measuring and Reducing Malicious Use With Unlearning](https://mlsanity.com/paper/2403.03218/) (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)
- [Scalable Extraction of Training Data from (Production) Language Models](https://mlsanity.com/paper/2311.17035/) (2023)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [SalUn: Empowering Machine Unlearning via Gradient-based Weight Saliency in Both Image Classification and Generation](https://mlsanity.com/paper/2310.12508/) (2023)
- [Who's Harry Potter? Approximate Unlearning in LLMs](https://mlsanity.com/paper/2310.02238/) (2023)
- [Remember What You Want to Forget: Algorithms for Machine Unlearning](https://mlsanity.com/paper/2103.03279/) (2021)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Large Language Model Unlearning](https://mlsanity.com/paper/2310.10683/) (2023)
- [TrustLLM: Trustworthiness in Large Language Models](https://mlsanity.com/paper/2401.05561/) (2024)
- [Unrolling SGD: Understanding Factors Influencing Machine Unlearning](https://mlsanity.com/paper/2109.13398/) (2021)
- [MUSE: Machine Unlearning Six-Way Evaluation for Language Models](https://mlsanity.com/paper/2407.06460/) (2024)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Gender bias and stereotypes in Large Language Models](https://mlsanity.com/paper/2308.14921/) (2023)
- [Simplicity Prevails: Rethinking Negative Preference Optimization for LLM Unlearning](https://mlsanity.com/paper/2410.07163/) (2024)
- [Defensive Unlearning with Adversarial Training for Robust Concept Erasure in Diffusion Models](https://mlsanity.com/paper/2405.15234/) (2024)
- [Eight Methods to Evaluate Robust Unlearning in LLMs](https://mlsanity.com/paper/2402.16835/) (2024)
- [Fairness in Machine Learning: Lessons from Political Philosophy](https://mlsanity.com/paper/1712.03586/) (2017)
- [Towards Safer Large Language Models through Machine Unlearning](https://mlsanity.com/paper/2402.10058/) (2024)
- [Graph Unlearning](https://mlsanity.com/paper/2103.14991/) (2021)
- [Continual Learning and Private Unlearning](https://mlsanity.com/paper/2203.12817/) (2022)
- [An Adversarial Perspective on Machine Unlearning for AI Safety](https://mlsanity.com/paper/2409.18025/) (2024)
- [Copyright Violations and Large Language Models](https://mlsanity.com/paper/2310.13771/) (2023)

## Built on by (most cited first, 23 of 23)

- [Leak@k: Unlearning Does Not Make LLMs Forget Under Probabilistic Decoding](https://mlsanity.com/paper/2511.04934/) (2025)
- [CURE:Circuit-Aware Unlearning for LLM-based Recommendation](https://mlsanity.com/paper/2604.04982/) (2026)
- [Downgrade to Upgrade: Optimizer Simplification Enhances Robustness in LLM Unlearning](https://mlsanity.com/paper/2510.00761/) (2025)
- [On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization](https://mlsanity.com/paper/2507.23155/) (2025)
- [STaR: Sensitive Trajectory Regulation for Unlearning in Large Reasoning Models](https://mlsanity.com/paper/2601.09281/) (2026)
- [A Mechanistic Perspective and Circuit-Guided Difficulty Metric for Unlearning](https://mlsanity.com/paper/2601.09624/) (2026)
- [MEMOREPAIR: Barrier-First Cascade Repair in Agentic Memory](https://mlsanity.com/paper/2605.07242/) (2026)
- [Inference-Time Machine Unlearning via Gated Activation Redirection](https://mlsanity.com/paper/2605.12765/) (2026)
- [TRACER: Token ReAssignment for Concept ERasure in Generative Recommendation](https://mlsanity.com/paper/2606.07688/) (2026)
- [Subspace Control: Turning Constrained Model Steering into Controllable Spectral Optimization](https://mlsanity.com/paper/2604.04231/) (2026)
- [Modeling LLM Unlearning as an Asymmetric Two-Task Learning Problem](https://mlsanity.com/paper/2604.14808/) (2026)
- [Unlearners Can Lie: Evaluating and Improving Honesty in LLM Unlearning](https://mlsanity.com/paper/2605.08765/) (2026)
- [TEMPO: Temporal Enforcement via Mode-Separated Policy Optimization for Trustworthy LLM Backtesting](https://mlsanity.com/paper/2605.18843/) (2026)
- [RapidUn: Influence-Driven Parameter Reweighting for Efficient Large Language Model Unlearning](https://mlsanity.com/paper/2512.04457/) (2025)
- [DualOptim+: Bridging Shared and Decoupled Optimizer States for Better Machine Unlearning in Large Language Models](https://mlsanity.com/paper/2605.21539/) (2026)
- [Replay What Matters: Off-Policy Replay for Efficient LLM Reinforcement Unlearning](https://mlsanity.com/paper/2606.15333/) (2026)
- [Understanding Machine Unlearning Through the Lens of Mode Connectivity](https://mlsanity.com/paper/2607.23970/) (2026)
- [SAUL: Sharpness-Aware Augmented-Lagrangian Unlearning](https://mlsanity.com/paper/2608.16249/) (2026)
- [BLADE: Bilevel Low-rank Augmented-Lagrangian Erasure for LLM Unlearning](https://mlsanity.com/paper/2608.22557/) (2026)
- [Stochastic Nonconvex Bilevel Optimization: Improved Rates Without Rare-Visit Assumption](https://mlsanity.com/paper/2609.06580/) (2026)
- [When Retain Constraints Conflict: Mitigating Forget-Retain Interference in Tabular Data](https://mlsanity.com/paper/2609.06786/) (2026)
- [Do LLMs Really Forget? Hidden-State Leakage in Model Unlearning and How to Fix it](https://mlsanity.com/paper/2609.36612/) (2026)
- [Faithful Dual-constrained Erasure for Robust LLM Safety Alignment](https://mlsanity.com/paper/2609.39279/) (2026)

## Abstract (arXiv)

Enabling large language models (LLMs) to unlearn knowledge and capabilities acquired during training has proven vital for ensuring compliance with data regulations and promoting ethical practices in generative AI. Although there are growing interests in developing various unlearning algorithms, it remains unclear how to best formulate the unlearning problem. The most popular formulation uses a weighted sum of forget and retain loss, but it often leads to performance degradation due to the inherent trade-off between forget and retain losses. In this work, we argue that it is important to model the hierarchical structure of the unlearning problem, where the forget problem (which textitunlearns certain knowledge and/or capabilities) takes priority over the retain problem (which preserves model utility). This hierarchical structure naturally leads to a bi-level optimization formulation where the lower-level objective focuses on minimizing the forget loss, while the upper-level objective aims to maintain the model's utility. Based on this new formulation, we propose a novel algorithm, termed Bi-Level UnleaRning (textttBLUR), which not only possesses strong theoretical guarantees but more importantly, delivers superior performance. In particular, our extensive experiments demonstrate that textttBLUR consistently outperforms all the state-of-the-art algorithms across various unlearning tasks, models, and metrics. Codes are available at https://github.com/OptimAI-Lab/BLURLLMUnlearning.