# Statistical Benefits of Fine-Tuning from Pretrained Initialization in Diagonal Linear Networks

arXiv: https://arxiv.org/abs/2609.34756 · HTML: https://mlsanity.com/paper/2609.34756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 37 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [Fine-Tuning can Distort Pretrained Features and Underperform Out-of-Distribution](https://mlsanity.com/paper/2202.10054/) (2022)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Do Better ImageNet Models Transfer Better?](https://mlsanity.com/paper/1805.08974/) (2018)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [A Kernel-Based View of Language Model Fine-Tuning](https://mlsanity.com/paper/2210.05643/) (2022)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Saddle-to-Saddle Dynamics in Diagonal Linear Networks](https://mlsanity.com/paper/2304.00488/) (2023)
- [Exponentiated Gradient Meets Gradient Descent](https://mlsanity.com/paper/1902.01903/) (2019)
- [Incremental Learning in Diagonal Linear Networks](https://mlsanity.com/paper/2208.14673/) (2022)
- [The Power and Limitation of Pretraining-Finetuning for Linear Regression under Covariate Shift](https://mlsanity.com/paper/2208.01857/) (2022)
- [High-Dimensional Linear Regression via Implicit Regularization](https://mlsanity.com/paper/1903.09367/) (2019)
- [Understanding Linear Probing then Fine-tuning Language Models from NTK Perspective](https://mlsanity.com/paper/2405.16747/) (2024)
- [The Statistical Complexity of Early-Stopped Mirror Descent](https://mlsanity.com/paper/2002.00189/) (2020)
- [A Theoretical Analysis of Fine-tuning with Linear Teachers](https://mlsanity.com/paper/2107.01641/) (2021)
- [Sparse Recovery via Differential Inclusions](https://mlsanity.com/paper/1406.7728/) (2014)
- [Inductive biases of multi-task learning and finetuning: multiple regimes of feature reuse](https://mlsanity.com/paper/2310.02396/) (2023)
- [Provable Benefits of Unsupervised Pre-training and Transfer Learning via Single-Index Models](https://mlsanity.com/paper/2502.16849/) (2025)
- [A Theory of How Pretraining Shapes Inductive Bias in Fine-Tuning](https://mlsanity.com/paper/2602.20062/) (2026)
- [Transfer Learning in Infinite Width Feature Learning Networks](https://mlsanity.com/paper/2507.04448/) (2025)
- [Incremental Learning in Mirror Flows](https://mlsanity.com/paper/2606.23198/) (2026)

## Abstract (arXiv)

Adapting pretrained models to downstream tasks with limited data has become a central paradigm in modern deep learning. Yet, despite its widespread practical success, how fine-tuning leverages information from pretraining remains poorly understood theoretically. We study fine-tuning from pretrained weights through the lens of sparse linear regression and two-layer diagonal linear networks. In our setting, pretraining provides information through the support (and signs) of the initialization predictor, which may contain coordinates relevant to the downstream task. We show how pretrained information reshapes the implicit bias and training dynamics, and can thereby reduce the sample complexity of recovering the target parameters and support. In particular, for a clean initialization with correctly inherited signs, we show that the required sample size is comparable to that of a weighted Lasso estimator that explicitly exploits the pretrained support through a suitably chosen regularizer. Our results thus show how information encoded in pretrained weights can be implicitly exploited by gradient-based fine-tuning, reducing the amount of data needed to recover a downstream task.