# Initialization using Update Approximation is a Silver Bullet for Extremely Efficient Low-Rank Fine-Tuning

arXiv: https://arxiv.org/abs/2411.19557 · HTML: https://mlsanity.com/paper/2411.19557/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 65 references that are in the map.
- First posted on arXiv 29 Nov 2024.

## Builds on (most cited first, 30 of 45)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Know What You Don't Know: Unanswerable Questions for SQuAD](https://mlsanity.com/paper/1806.03822/) (2018)
- [SemEval-2017 Task 1: Semantic Textual Similarity - Multilingual and Cross-lingual Focused Evaluation](https://mlsanity.com/paper/1708.00055/) (2017)
- [Neural Network Acceptability Judgments](https://mlsanity.com/paper/1805.12471/) (2018)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [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)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)

## Built on by (most cited first, 22 of 22)

- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [Beyond Zero Initialization: Investigating the Impact of Non-Zero Initialization on LoRA Fine-Tuning Dynamics](https://mlsanity.com/paper/2505.23194/) (2025)
- [Find Your Optimal Teacher: Personalized Data Synthesis via Router-Guided Multi-Teacher Distillation](https://mlsanity.com/paper/2510.10925/) (2025)
- [LoFT: Low-Rank Adaptation That Behaves Like Full Fine-Tuning](https://mlsanity.com/paper/2505.21289/) (2025)
- [S2FT: Efficient, Scalable and Generalizable LLM Fine-tuning by Structured Sparsity](https://mlsanity.com/paper/2412.06289/) (2024)
- [Fed-SB: A Silver Bullet for Extreme Communication Efficiency and Performance in (Private) Federated LoRA Fine-Tuning](https://mlsanity.com/paper/2502.15436/) (2025)
- [GraLoRA: Granular Low-Rank Adaptation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2505.20355/) (2025)
- [AltLoRA: Towards Better Gradient Approximation in Low-Rank Adaptation with Alternating Projections](https://mlsanity.com/paper/2505.12455/) (2025)
- [Ravan: Multi-Head Low-Rank Adaptation for Federated Fine-Tuning](https://mlsanity.com/paper/2506.05568/) (2025)
- [Beyond Higher Rank: Token-wise Input-Output Projections for Efficient Low-Rank Adaptation](https://mlsanity.com/paper/2510.23123/) (2025)
- [ABBA-Adapters: Efficient and Expressive Fine-Tuning of Foundation Models](https://mlsanity.com/paper/2505.14238/) (2025)
- [DiaBlo: Diagonal Blocks Are Sufficient For Finetuning](https://mlsanity.com/paper/2506.03230/) (2025)
- [A Unified Study of LoRA Variants: Taxonomy, Review, Codebase, and Empirical Evaluation](https://mlsanity.com/paper/2601.22708/) (2026)
- [HRP: High-Rank Preheating for Superior LoRA Initialization](https://mlsanity.com/paper/2502.07739/) (2025)
- [Beyond LoRA: Is Sparsity-Induced Adaptation Better?](https://mlsanity.com/paper/2606.13767/) (2026)
- [GiVA: Gradient-Informed Bases for Vector-Based Adaptation](https://mlsanity.com/paper/2604.21901/) (2026)
- [UB-SMoE: Universally Balanced Sparse Mixture-of-Experts for Resource-adaptive Federated Fine-tuning of Foundation Models](https://mlsanity.com/paper/2605.16690/) (2026)
- [Principal-timestep Restricted Init via Sparse Matrix-decomposition in Flow-matching](https://mlsanity.com/paper/2609.15643/) (2026)
- [Adaptive Utilization of Low-Rank Adaptation via Conditioned Gating](https://mlsanity.com/paper/2610.05800/) (2026)
- [Frozen Factor or Spectral Band? Disentangling Two Choices in Low-Rank LoRA](https://mlsanity.com/paper/2610.06621/) (2026)
- [Are Parameter-Efficient Fine-tuning Methods Really Different?](https://mlsanity.com/paper/2610.09122/) (2026)
- [ReGRA: Residual-Gradient Adaptation for Complementary Low-Rank Fine-Tuning](https://mlsanity.com/paper/2610.19700/) (2026)

## Abstract (arXiv)

Low-rank adapters have become standard for efficiently fine-tuning large language models, but they often fall short of achieving the performance of full fine-tuning. We propose a method, LoRA Silver Bullet or LoRA-SB, that approximates full fine-tuning within low-rank subspaces using a carefully designed initialization strategy. We theoretically demonstrate that the architecture of LoRA-XS, which inserts a learnable r x r matrix between B and A while keeping other matrices fixed, provides the precise conditions needed for this approximation. We leverage its constrained update space to achieve optimal scaling for high-rank gradient updates while removing the need for scaling factor tuning. We prove that our initialization offers an optimal low-rank approximation of the initial gradient and preserves update directions throughout training. Extensive experiments across mathematical reasoning, commonsense reasoning, and language understanding tasks demonstrate that our approach exceeds the performance of LoRA (and baselines) while using 27-90 times fewer learnable parameters, and comprehensively outperforms LoRA-XS. Our findings establish that it is possible to simulate full fine-tuning in low-rank subspaces, and achieve significant parameter efficiency gains without sacrificing performance. Our code is publicly available at: https://github.com/CERT-Lab/lora-sb.