# Learning and generalization of one-hidden-layer neural networks, going beyond standard Gaussian data

arXiv: https://arxiv.org/abs/2207.03615 · HTML: https://mlsanity.com/paper/2207.03615/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 54 references that are in the map.
- First posted on arXiv 7 Jul 2022.

## Builds on (most cited first, 30 of 37)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (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)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (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)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Free Lunch for Few-shot Learning: Distribution Calibration](https://mlsanity.com/paper/2101.06395/) (2021)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Theoretical insights into the optimization landscape of over-parameterized shallow neural networks](https://mlsanity.com/paper/1707.04926/) (2017)
- [An Improved Analysis of Training Over-parameterized Deep Neural Networks](https://mlsanity.com/paper/1906.04688/) (2019)
- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [On the Computational Efficiency of Training Neural Networks](https://mlsanity.com/paper/1410.1141/) (2014)
- [Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification](https://mlsanity.com/paper/2006.06098/) (2020)
- [Mean Field Limit of the Learning Dynamics of Multilayer Neural Networks](https://mlsanity.com/paper/1902.02880/) (2019)
- [Fast Learning of Graph Neural Networks with Guaranteed Generalizability: One-hidden-layer Case](https://mlsanity.com/paper/2006.14117/) (2020)
- [Data-Dependence of Plateau Phenomenon in Learning with Neural Network --- Statistical Mechanical Analysis](https://mlsanity.com/paper/2001.03371/) (2020)
- [Guaranteed Recovery of One-Hidden-Layer Neural Networks via Cross Entropy](https://mlsanity.com/paper/1802.06463/) (2018)

## Built on by (most cited first, 8 of 8)

- [A Theoretical Understanding of Shallow Vision Transformers: Learning, Generalization, and Sample Complexity](https://mlsanity.com/paper/2302.06015/) (2023)
- [Patch-level Routing in Mixture-of-Experts is Provably Sample-efficient for Convolutional Neural Networks](https://mlsanity.com/paper/2306.04073/) (2023)
- [How Do Nonlinear Transformers Learn and Generalize in In-Context Learning?](https://mlsanity.com/paper/2402.15607/) (2024)
- [What Improves the Generalization of Graph Transformers? A Theoretical Dive into the Self-attention and Positional Encoding](https://mlsanity.com/paper/2406.01977/) (2024)
- [Learning on Transformers is Provable Low-Rank and Sparse: A One-layer Analysis](https://mlsanity.com/paper/2406.17167/) (2024)
- [How does promoting the minority fraction affect generalization? A theoretical study of the one-hidden-layer neural network on group imbalance](https://mlsanity.com/paper/2403.07310/) (2024)
- [How Can Mamba Learn In Context with Outliers and Generalize Provably?](https://mlsanity.com/paper/2510.00399/) (2025)
- [Theoretical Learning Performance of Graph Neural Networks: The Impact of Jumping Connections and Layer-wise Sparsification](https://mlsanity.com/paper/2507.05533/) (2025)

## Abstract (arXiv)

This paper analyzes the convergence and generalization of training a one-hidden-layer neural network when the input features follow the Gaussian mixture model consisting of a finite number of Gaussian distributions. Assuming the labels are generated from a teacher model with an unknown ground truth weight, the learning problem is to estimate the underlying teacher model by minimizing a non-convex risk function over a student neural network. With a finite number of training samples, referred to the sample complexity, the iterations are proved to converge linearly to a critical point with guaranteed generalization error. In addition, for the first time, this paper characterizes the impact of the input distributions on the sample complexity and the learning rate.