# On the Cone Effect in the Learning Dynamics

arXiv: https://arxiv.org/abs/2503.16316 · HTML: https://mlsanity.com/paper/2503.16316/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 33 references that are in the map.
- First posted on arXiv 20 Mar 2025.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [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 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)
- [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)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (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)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)
- [Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity](https://mlsanity.com/paper/2307.08286/) (2023)
- [From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks](https://mlsanity.com/paper/2409.14623/) (2024)
- [Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning](https://mlsanity.com/paper/2406.06158/) (2024)
- [On the training dynamics of deep networks with L2 regularization](https://mlsanity.com/paper/2006.08643/) (2020)
- [Feature Learning in L2-regularized DNNs: Attraction/Repulsion and Sparsity](https://mlsanity.com/paper/2205.15809/) (2022)
- [How connectivity structure shapes rich and lazy learning in neural circuits](https://mlsanity.com/paper/2310.08513/) (2023)

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

- [Mapping the Emergence of Regularization-Driven Dynamics in Grokking](https://mlsanity.com/paper/2608.25813/) (2026)
- [Reactivation: Empirical NTK Dynamics Under Task Shifts](https://mlsanity.com/paper/2507.16039/) (2025)

## Abstract (arXiv)

Understanding the learning dynamics of neural networks is a central topic in the deep learning community. In this paper, we take an empirical perspective to study the learning dynamics of neural networks in real-world settings. Specifically, we investigate the evolution process of the empirical Neural Tangent Kernel (eNTK) during training. Our key findings reveal a two-phase learning process: i) in Phase I, the eNTK evolves significantly, signaling the rich regime, and ii) in Phase II, the eNTK keeps evolving but is constrained in a narrow space, a phenomenon we term the cone effect. This two-phase framework builds on the hypothesis proposed by Fort et al. (2020), but we uniquely identify the cone effect in Phase II, demonstrating its significant performance advantages over fully linearized training.