# Norm-Bounded Low-Rank Adaptation

arXiv: https://arxiv.org/abs/2501.19050 · HTML: https://mlsanity.com/paper/2501.19050/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 45 references that are in the map.
- First posted on arXiv 31 Jan 2025.

## Builds on (most cited first, 29 of 29)

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- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [VeRA: Vector-based Random Matrix Adaptation](https://mlsanity.com/paper/2310.11454/) (2023)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [OpenCodeInterpreter: Integrating Code Generation with Execution and Refinement](https://mlsanity.com/paper/2402.14658/) (2024)
- [Controlling Text-to-Image Diffusion by Orthogonal Finetuning](https://mlsanity.com/paper/2306.07280/) (2023)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [LoRA-XS: Low-Rank Adaptation with Extremely Small Number of Parameters](https://mlsanity.com/paper/2405.17604/) (2024)
- [Parameter-Efficient Fine-Tuning Design Spaces](https://mlsanity.com/paper/2301.01821/) (2023)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [The Impact of Initialization on LoRA Finetuning Dynamics](https://mlsanity.com/paper/2406.08447/) (2024)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Distance-Based Regularisation of Deep Networks for Fine-Tuning](https://mlsanity.com/paper/2002.08253/) (2020)
- [LoRA Training in the NTK Regime has No Spurious Local Minima](https://mlsanity.com/paper/2402.11867/) (2024)
- [Computational Limits of Low-Rank Adaptation (LoRA) Fine-Tuning for Transformer Models](https://mlsanity.com/paper/2406.03136/) (2024)
- [Parameter Efficient Fine-tuning of Self-supervised ViTs without Catastrophic Forgetting](https://mlsanity.com/paper/2404.17245/) (2024)
- [ETHER: Efficient Finetuning of Large-Scale Models with Hyperplane Reflections](https://mlsanity.com/paper/2405.20271/) (2024)

## Abstract (arXiv)

In this work, we propose norm-bounded low-rank adaptation (NB-LoRA) for parameter-efficient fine tuning. NB-LoRA is a novel parameterization of low-rank weight adaptations that admits explicit bounds on each singular value of the adaptation matrix, which can thereby satisfy any prescribed unitarily invariant norm bound, including the Schatten norms (e.g., nuclear, Frobenius, spectral norm). The proposed parameterization is unconstrained, smooth, and complete, i.e. it covers all matrices satisfying the prescribed rank and singular-value bounds. Natural language generation experiments show that NB-LoRA matches or surpasses performance of competing LoRA methods, while exhibiting stronger hyper-parameter robustness. Vision fine-tuning experiments show that NB-LoRA can avoid model catastrophic forgetting without minor cost on adaptation performance, and compared to existing approaches it is substantially more robust to a hyper-parameters such as including adaptation rank, learning rate and number of training epochs.