# The lazy (NTK) and rich (μP) regimes: a gentle tutorial

arXiv: https://arxiv.org/abs/2404.19719 · HTML: https://mlsanity.com/paper/2404.19719/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 23 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 20 of 20)

- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [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 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)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [Tensor Programs II: Neural Tangent Kernel for Any Architecture](https://mlsanity.com/paper/2006.14548/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks](https://mlsanity.com/paper/2205.09653/) (2022)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [Neural Kernels Without Tangents](https://mlsanity.com/paper/2003.02237/) (2020)
- [Feature-Learning Networks Are Consistent Across Widths At Realistic Scales](https://mlsanity.com/paper/2305.18411/) (2023)
- [The Eigenlearning Framework: A Conservation Law Perspective on Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2110.03922/) (2021)
- [A Theory of Neural Tangent Kernel Alignment and Its Influence on Training](https://mlsanity.com/paper/2105.14301/) (2021)
- [More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory](https://mlsanity.com/paper/2311.14646/) (2023)

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

- [Rethinking Conventional Wisdom in Machine Learning: From Generalization to Scaling](https://mlsanity.com/paper/2409.15156/) (2024)
- [K"ahler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold](https://mlsanity.com/paper/2608.19584/) (2026)

## Abstract (arXiv)

A central theme of the modern machine learning paradigm is that larger neural networks achieve better performance on a variety of metrics. Theoretical analyses of these overparameterized models have recently centered around studying very wide neural networks. In this tutorial, we provide a nonrigorous but illustrative derivation of the following fact: in order to train wide networks effectively, there is only one degree of freedom in choosing hyperparameters such as the learning rate and the size of the initial weights. This degree of freedom controls the richness of training behavior: at minimum, the wide network trains lazily like a kernel machine, and at maximum, it exhibits feature learning in the active μP regime. In this paper, we explain this richness scale, synthesize recent research results into a coherent whole, offer new perspectives and intuitions, and provide empirical evidence supporting our claims. In doing so, we hope to encourage further study of the richness scale, as it may be key to developing a scientific theory of feature learning in practical deep neural networks.