# High-Dimensional Theory of LoRA Fine-Tuning in a Solvable Attention Model

arXiv: https://arxiv.org/abs/2606.05899 · HTML: https://mlsanity.com/paper/2606.05899/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 38 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 30 of 33)

- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [A Rank Stabilization Scaling Factor for Fine-Tuning with LoRA](https://mlsanity.com/paper/2312.03732/) (2023)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [The Expressive Power of Low-Rank Adaptation](https://mlsanity.com/paper/2310.17513/) (2023)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Learning time-scales in two-layers neural networks](https://mlsanity.com/paper/2303.00055/) (2023)
- [LoRA Training in the NTK Regime has No Spurious Local Minima](https://mlsanity.com/paper/2402.11867/) (2024)
- [LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently](https://mlsanity.com/paper/2502.01235/) (2025)
- [A phase transition between positional and semantic learning in a solvable model of dot-product attention](https://mlsanity.com/paper/2402.03902/) (2024)
- [Universality laws for Gaussian mixtures in generalized linear models](https://mlsanity.com/paper/2302.08933/) (2023)
- [An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings](https://mlsanity.com/paper/2205.06308/) (2022)
- [LoRA Training Provably Converges to a Low-Rank Global Minimum or It Fails Loudly (But it Probably Won't Fail)](https://mlsanity.com/paper/2502.09376/) (2025)
- [Asymptotics of Non-Convex Generalized Linear Models in High-Dimensions: A proof of the replica formula](https://mlsanity.com/paper/2502.20003/) (2025)
- [Understanding the Learning Dynamics of LoRA: A Gradient Flow Perspective on Low-Rank Adaptation in Matrix Factorization](https://mlsanity.com/paper/2503.06982/) (2025)
- [High-dimensional learning of narrow neural networks](https://mlsanity.com/paper/2409.13904/) (2024)
- [Fundamental limits of learning in sequence multi-index models and deep attention networks: High-dimensional asymptotics and sharp thresholds](https://mlsanity.com/paper/2502.00901/) (2025)
- [Fundamental Limits of Matrix Sensing: Exact Asymptotics, Universality, and Applications](https://mlsanity.com/paper/2503.14121/) (2025)
- [The Nuclear Route: Sharp Asymptotics of ERM in Overparameterized Quadratic Networks](https://mlsanity.com/paper/2505.17958/) (2025)
- [Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapters](https://mlsanity.com/paper/2506.14530/) (2025)
- [DeLoRA: Decoupling Angles and Strength in Low-rank Adaptation](https://mlsanity.com/paper/2503.18225/) (2025)
- [On the Convergence Rate of LoRA Gradient Descent](https://mlsanity.com/paper/2512.18248/) (2025)
- [Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks](https://mlsanity.com/paper/2506.02651/) (2025)
- [A solvable high-dimensional model where nonlinear autoencoders learn structure invisible to PCA while test loss misaligns with generalization](https://mlsanity.com/paper/2602.10680/) (2026)
- [Statistical Advantage of Softmax Attention: Insights from Single-Location Regression](https://mlsanity.com/paper/2509.21936/) (2025)
- [Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws](https://mlsanity.com/paper/2509.24914/) (2025)
- [Does Low Rank Adaptation Lead to Lower Robustness against Training-Time Attacks?](https://mlsanity.com/paper/2505.12871/) (2025)

## Built on by (most cited first, 3 of 3)

- [The Intruder Threshold: A Spectral Law for LoRA Fine-Tuning](https://mlsanity.com/paper/2607.23711/) (2026)
- [How Much Rank Does LoRA Need? Rank-Error Bounds for Transformer Attention](https://mlsanity.com/paper/2608.26052/) (2026)
- [A Dynamical Theory of LoRA in Continual Learning](https://mlsanity.com/paper/2609.39367/) (2026)

## Abstract (arXiv)

We develop a high-dimensional statistical theory of low-rank adaptation (LoRA) in attention models, capturing the interplay between pre-training and fine-tuning. We introduce a solvable framework in which a single-head attention layer is first pre-trained on a data-abundant task and subsequently adapted via a rank-one LoRA update on limited data. In the high-dimensional limit, both stages admit a sharp asymptotic characterization in terms of a finite set of order parameters, yielding explicit predictions for test errors and representation alignment. Our analysis shows that the impact of pre-training on LoRA is summarized by an effective noise term, from which we derive prescriptions for the optimal pre-training procedure. We also demonstrate a regime with a mismatch between the value of the test error and representation quality, and propose an application of our theory to active fine-tuning.