# Provable Guarantees for Neural Networks via Gradient Feature Learning

arXiv: https://arxiv.org/abs/2310.12408 · HTML: https://mlsanity.com/paper/2310.12408/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 124 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Builds on (most cited first, 30 of 63)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [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)
- [The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks](https://mlsanity.com/paper/2202.08658/) (2022)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Finite Depth and Width Corrections to the Neural Tangent Kernel](https://mlsanity.com/paper/1909.05989/) (2019)
- [Dynamics of Deep Neural Networks and Neural Tangent Hierarchy](https://mlsanity.com/paper/1909.08156/) (2019)
- [Are All Layers Created Equal?](https://mlsanity.com/paper/1902.01996/) (2019)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [What Can ResNet Learn Efficiently, Going Beyond Kernels?](https://mlsanity.com/paper/1905.10337/) (2019)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1709.01953/) (2017)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Beyond Linearization: On Quadratic and Higher-Order Approximation of Wide Neural Networks](https://mlsanity.com/paper/1910.01619/) (2019)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)

## Built on by (most cited first, 14 of 14)

- [Why Larger Language Models Do In-context Learning Differently?](https://mlsanity.com/paper/2405.19592/) (2024)
- [Towards Few-Shot Adaptation of Foundation Models via Multitask Finetuning](https://mlsanity.com/paper/2402.15017/) (2024)
- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [BiSHop: Bi-Directional Cellular Learning for Tabular Data with Generalized Sparse Modern Hopfield Model](https://mlsanity.com/paper/2404.03830/) (2024)
- [Looped ReLU MLPs May Be All You Need as Practical Programmable Computers](https://mlsanity.com/paper/2410.09375/) (2024)
- [Differential Privacy Mechanisms in Neural Tangent Kernel Regression](https://mlsanity.com/paper/2407.13621/) (2024)
- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [Guided by the Experts: Provable Feature Learning Dynamic of Soft-Routed Mixture-of-Experts](https://mlsanity.com/paper/2510.07205/) (2025)
- [Unifying Learning Dynamics and Generalization in Transformers Scaling Law](https://mlsanity.com/paper/2512.22088/) (2025)
- [Robust Fourier Neural Networks](https://mlsanity.com/paper/2409.02052/) (2024)
- [Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networks](https://mlsanity.com/paper/2409.19138/) (2024)
- [Theoretical Foundations of Representation Learning using Unlabeled Data: Statistics and Optimization](https://mlsanity.com/paper/2509.18997/) (2025)
- [Neuron Block Dynamics for XOR Classification with Zero-Margin](https://mlsanity.com/paper/2602.00172/) (2026)
- [Training Dynamics of Transformers on Unbounded Inputs: Attention Learning Can Lag Output Layer Learning](https://mlsanity.com/paper/2610.20337/) (2026)

## Abstract (arXiv)

Neural networks have achieved remarkable empirical performance, while the current theoretical analysis is not adequate for understanding their success, e.g., the Neural Tangent Kernel approach fails to capture their key feature learning ability, while recent analyses on feature learning are typically problem-specific. This work proposes a unified analysis framework for two-layer networks trained by gradient descent. The framework is centered around the principle of feature learning from gradients, and its effectiveness is demonstrated by applications in several prototypical problems, such as mixtures of Gaussians and parity functions. The framework also sheds light on interesting network learning phenomena such as feature learning beyond kernels and the lottery ticket hypothesis.