# Low-rank bias, weight decay, and model merging in neural networks

arXiv: https://arxiv.org/abs/2502.17340 · HTML: https://mlsanity.com/paper/2502.17340/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 48 references that are in the map.
- First posted on arXiv 24 Feb 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Rewarded soups: towards Pareto-optimal alignment by interpolating weights fine-tuned on diverse rewards](https://mlsanity.com/paper/2306.04488/) (2023)
- [Patching open-vocabulary models by interpolating weights](https://mlsanity.com/paper/2208.05592/) (2022)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Diverse Weight Averaging for Out-of-Distribution Generalization](https://mlsanity.com/paper/2205.09739/) (2022)
- [WARM: On the Benefits of Weight Averaged Reward Models](https://mlsanity.com/paper/2401.12187/) (2024)
- [Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel](https://mlsanity.com/paper/1810.05369/) (2018)
- [Polylogarithmic width suffices for gradient descent to achieve arbitrarily small test error with shallow ReLU networks](https://mlsanity.com/paper/1909.12292/) (2019)
- [DiLoCo: Distributed Low-Communication Training of Language Models](https://mlsanity.com/paper/2311.08105/) (2023)
- [Inducing Neural Collapse in Imbalanced Learning: Do We Really Need a Learnable Classifier at the End of Deep Neural Network?](https://mlsanity.com/paper/2203.09081/) (2022)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)

## Built on by (most cited first, 5 of 5)

- [Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction](https://mlsanity.com/paper/2606.05863/) (2026)
- [Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks](https://mlsanity.com/paper/2608.06597/) (2026)
- [Manifold Projection and Iterative Autoencoder Refinement for Masked Language Modeling](https://mlsanity.com/paper/2609.30288/) (2026)
- [Learning Functional Subspaces for Neural Network Compression](https://mlsanity.com/paper/2609.40127/) (2026)
- [Weight Decay and Neuron Condensation: A Three-Stage Analysis of Two-Layer ReLU Networks](https://mlsanity.com/paper/2610.04533/) (2026)

## Abstract (arXiv)

We explore the low-rank structure of the weight matrices in neural networks at the stationary points (limiting solutions of optimization algorithms) with L2 regularization (also known as weight decay). We show several properties of such deep neural networks, induced by L2 regularization. In particular, for a stationary point we show alignment of the parameters and the gradient, norm preservation across layers, and low-rank bias: properties previously known in the context of solution of gradient descent/flow type algorithms. Experiments show that the assumptions made in the analysis only mildly affect the observations. In addition, we investigate a multitask learning phenomenon enabled by L2 regularization and low-rank bias. In particular, we show that if two networks are trained, such that the inputs in the training set of one network are approximately orthogonal to the inputs in the training set of the other network, the new network obtained by simply summing the weights of the two networks will perform as well on both training sets as the respective individual networks. We demonstrate this for shallow ReLU neural networks trained by gradient descent, as well as deep linear networks trained by gradient flow.