# From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks

arXiv: https://arxiv.org/abs/2409.14623 · HTML: https://mlsanity.com/paper/2409.14623/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 85 references that are in the map.
- First posted on arXiv 22 Sep 2024.

## Builds on (most cited first, 30 of 54)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Relative representations enable zero-shot latent space communication](https://mlsanity.com/paper/2209.15430/) (2022)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)

## Built on by (most cited first, 30 of 49)

- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer](https://mlsanity.com/paper/2502.02531/) (2025)
- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures](https://mlsanity.com/paper/2512.20607/) (2025)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Bridging Critical Gaps in Convergent Learning: How Representational Alignment Evolves Across Layers, Training, and Distribution Shifts](https://mlsanity.com/paper/2502.18710/) (2025)
- [A Theory of Initialisation's Impact on Specialisation](https://mlsanity.com/paper/2503.02526/) (2025)
- [Sign-In to the Lottery: Reparameterizing Sparse Training From Scratch](https://mlsanity.com/paper/2504.12801/) (2025)
- [Path-conditioned training: a principled way to rescale ReLU neural networks](https://mlsanity.com/paper/2602.19799/) (2026)
- [Spectral Dynamics in Deep Networks: Feature Learning, Outlier Escape, and Learning Rate Transfer](https://mlsanity.com/paper/2605.07870/) (2026)
- [A Theory of How Pretraining Shapes Inductive Bias in Fine-Tuning](https://mlsanity.com/paper/2602.20062/) (2026)
- [Intrinsic training dynamics of deep neural networks](https://mlsanity.com/paper/2508.07370/) (2025)
- [Never Saddle for Reparameterized Steepest Descent as Mirror Flow](https://mlsanity.com/paper/2603.02064/) (2026)
- [From Density Matrices to Phase Transitions in Deep Learning: Spectral Early Warnings and Interpretability](https://mlsanity.com/paper/2603.29805/) (2026)
- [You Had One Job: Per-Task Quantization Using LLMs' Hidden Representations](https://mlsanity.com/paper/2511.06516/) (2025)
- [A Generalized Spectral Framework to Expain Neural Scaling and Compression Dynamics](https://mlsanity.com/paper/2511.07892/) (2025)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [Decoupling Dynamical Richness from Representation Learning: Towards Practical Measurement](https://mlsanity.com/paper/2410.04264/) (2024)
- [On the Cone Effect in the Learning Dynamics](https://mlsanity.com/paper/2503.16316/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [The Impact of Anisotropic Covariance Structure on the Training Dynamics and Generalization Error of Linear Networks](https://mlsanity.com/paper/2601.06961/) (2026)
- [How Controlling the Variance can Improve Training Stability of Sparsely Activated DNNs and CNNs](https://mlsanity.com/paper/2602.05779/) (2026)
- [Error Bound Analysis for the Regularized Loss of Deep Linear Neural Networks](https://mlsanity.com/paper/2502.11152/) (2025)
- [Feature learning is decoupled from generalization in high capacity neural networks](https://mlsanity.com/paper/2507.19680/) (2025)
- [Tuning the Implicit Regularizer of Masked Diffusion Language Models: Enhancing Generalization via Insights from k-Parity](https://mlsanity.com/paper/2601.22450/) (2026)
- [Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutions](https://mlsanity.com/paper/2602.04548/) (2026)
- [Not all solutions are created equal: An analytical dissociation of functional and representational similarity in deep linear neural networks](https://mlsanity.com/paper/2609.38998/) (2026)
- [Data Curation Through the Lens of Spectral Dynamics: Static Limits, Dynamic Acceleration, and Practical Oracles](https://mlsanity.com/paper/2512.02409/) (2025)

## Abstract (arXiv)

Biological and artificial neural networks develop internal representations that enable them to perform complex tasks. In artificial networks, the effectiveness of these models relies on their ability to build task specific representation, a process influenced by interactions among datasets, architectures, initialization strategies, and optimization algorithms. Prior studies highlight that different initializations can place networks in either a lazy regime, where representations remain static, or a rich/feature learning regime, where representations evolve dynamically. Here, we examine how initialization influences learning dynamics in deep linear neural networks, deriving exact solutions for lambda-balanced initializations-defined by the relative scale of weights across layers. These solutions capture the evolution of representations and the Neural Tangent Kernel across the spectrum from the rich to the lazy regimes. Our findings deepen the theoretical understanding of the impact of weight initialization on learning regimes, with implications for continual learning, reversal learning, and transfer learning, relevant to both neuroscience and practical applications.