# Foundation-Preserving Optimization in Generalized Eigenspace

arXiv: https://arxiv.org/abs/2606.00132 · HTML: https://mlsanity.com/paper/2606.00132/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 49 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 30 of 40)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Fine-tuning Aligned Language Models Compromises Safety, Even When Users Do Not Intend To!](https://mlsanity.com/paper/2310.03693/) (2023)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution](https://mlsanity.com/paper/2202.10054/) (2022)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [GLM-5: from Vibe Coding to Agentic Engineering](https://mlsanity.com/paper/2602.15763/) (2026)
- [Kimi K2.5: Visual Agentic Intelligence](https://mlsanity.com/paper/2602.02276/) (2026)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [LongBench v2: Towards Deeper Understanding and Reasoning on Realistic Long-context Multitasks](https://mlsanity.com/paper/2412.15204/) (2024)
- [OpenCodeInterpreter: Integrating Code Generation with Execution and Refinement](https://mlsanity.com/paper/2402.14658/) (2024)
- [Self-Distillation Enables Continual Learning](https://mlsanity.com/paper/2601.19897/) (2026)
- [BABILong: Testing the Limits of LLMs with Long Context Reasoning-in-a-Haystack](https://mlsanity.com/paper/2406.10149/) (2024)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Understanding Catastrophic Forgetting in Language Models via Implicit Inference](https://mlsanity.com/paper/2309.10105/) (2023)
- [Revisiting Catastrophic Forgetting in Large Language Model Tuning](https://mlsanity.com/paper/2406.04836/) (2024)

## Abstract (arXiv)

While finetuning effectively adapts foundation models to specialized downstream tasks, it can degrade nontarget capabilities acquired during pretraining. Existing forgetting aware methods typically seek safer updates through specialized initialization or fixed constraints, but do not regulate the adaptation preservation trade-off during training. We propose Foundation Preserving LoRA (FoLoRA), a forgetting aware optimization framework. Guided by a first order preservation condition, FoLoRA defines a forgetting penalty over pretraining-proxy activations and a task utility over downstream task activations. It then scores update directions by task utility per unit forgetting penalty via a generalized Rayleigh quotient. The resulting spectral coordinate system enables direction wise gated Adam updates, attenuating low utility to penalty directions during training. To estimate the forgetting penalty, FoLoRA constructs pretraining proxy calibration data by sampling from the pretrained model rather than relying on a single proxy dataset. Experiments on math, code, and instruction following adaptation show that FoLoRA achieves the strongest preservation adaptation balance over baselines, improving target task performance with best aggregate preservation of non target capabilities.