# Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory

arXiv: https://arxiv.org/abs/2510.01930 · HTML: https://mlsanity.com/paper/2510.01930/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 59 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 30 of 36)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks](https://mlsanity.com/paper/2205.09653/) (2022)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [Shape Matters: Understanding the Implicit Bias of the Noise Covariance](https://mlsanity.com/paper/2006.08680/) (2020)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Implicit Bias in Deep Linear Classification: Initialization Scale vs Training Accuracy](https://mlsanity.com/paper/2007.06738/) (2020)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification](https://mlsanity.com/paper/2006.06098/) (2020)
- [Saddle-to-Saddle Dynamics in Diagonal Linear Networks](https://mlsanity.com/paper/2304.00488/) (2023)
- [A unifying tutorial on Approximate Message Passing](https://mlsanity.com/paper/2105.02180/) (2021)
- [Infinite Limits of Multi-head Transformer Dynamics](https://mlsanity.com/paper/2405.15712/) (2024)
- [(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability](https://mlsanity.com/paper/2302.08982/) (2023)

## Built on by (most cited first, 2 of 2)

- [Dichotomy of Feature Learning and Unlearning: Fast-Slow Analysis on Neural Networks with Stochastic Gradient Descent](https://mlsanity.com/paper/2602.07378/) (2026)
- [Statistical Vignettes on the Grokking Phenomenon](https://mlsanity.com/paper/2610.18811/) (2026)

## Abstract (arXiv)

Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training, such as initialization-dependent solutions and incremental learning. These phenomena are typically studied in isolation, leaving the overall dynamics insufficiently understood. In this work, we present a unified analysis of various phenomena in the gradient flow dynamics of DLNs. Using Dynamical Mean-Field Theory (DMFT), we derive a low-dimensional effective process that captures the asymptotic gradient flow dynamics in high dimensions. Analyzing this effective process yields new insights into DLN dynamics, including loss convergence rates and their trade-off with generalization, and systematically reproduces many of the previously observed phenomena. These findings deepen our understanding of DLNs and demonstrate the effectiveness of the DMFT approach in analyzing high-dimensional learning dynamics of neural networks.