# Benign Overfitting in Two-layer Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2202.06526 · HTML: https://mlsanity.com/paper/2202.06526/

## Key facts

- Cited by 91 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 71 references that are in the map.
- First posted on arXiv 14 Feb 2022.

## Builds on (most cited first, 30 of 56)

- [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)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (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)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [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)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning](https://mlsanity.com/paper/2012.09816/) (2020)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networks](https://mlsanity.com/paper/1907.04595/) (2019)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)

## Built on by (most cited first, 30 of 91)

- [Emergent and Predictable Memorization in Large Language Models](https://mlsanity.com/paper/2304.11158/) (2023)
- [Deep Learning is Not So Mysterious or Different](https://mlsanity.com/paper/2503.02113/) (2025)
- [Towards Understanding Mixture of Experts in Deep Learning](https://mlsanity.com/paper/2208.02813/) (2022)
- [On the Role of Attention in Prompt-tuning](https://mlsanity.com/paper/2306.03435/) (2023)
- [Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity](https://mlsanity.com/paper/2307.08286/) (2023)
- [Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data](https://mlsanity.com/paper/2210.07082/) (2022)
- [Robust Learning with Progressive Data Expansion Against Spurious Correlation](https://mlsanity.com/paper/2306.04949/) (2023)
- [When is Task Vector Provably Effective for Model Editing? A Generalization Analysis of Nonlinear Transformers](https://mlsanity.com/paper/2504.10957/) (2025)
- [Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting](https://mlsanity.com/paper/2207.06569/) (2022)
- [Understanding and Improving Feature Learning for Out-of-Distribution Generalization](https://mlsanity.com/paper/2304.11327/) (2023)
- [Benign Overfitting for Two-layer ReLU Convolutional Neural Networks](https://mlsanity.com/paper/2303.04145/) (2023)
- [Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data](https://mlsanity.com/paper/2310.02541/) (2023)
- [Implicit Bias and Fast Convergence Rates for Self-attention](https://mlsanity.com/paper/2402.05738/) (2024)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023)
- [The Double-Edged Sword of Implicit Bias: Generalization vs. Robustness in ReLU Networks](https://mlsanity.com/paper/2303.01456/) (2023)
- [Federated Learning from Vision-Language Foundation Models: Theoretical Analysis and Method](https://mlsanity.com/paper/2409.19610/) (2024)
- [Generalization and Stability of Interpolating Neural Networks with Minimal Width](https://mlsanity.com/paper/2302.09235/) (2023)
- [The Benefits of Mixup for Feature Learning](https://mlsanity.com/paper/2303.08433/) (2023)
- [Implicit Bias of Gradient Descent for Two-layer ReLU and Leaky ReLU Networks on Nearly-orthogonal Data](https://mlsanity.com/paper/2310.18935/) (2023)
- [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023)
- [Memorization in deep learning: A survey](https://mlsanity.com/paper/2406.03880/) (2024)
- [Wide Neural Networks as Gaussian Processes: Lessons from Deep Equilibrium Models](https://mlsanity.com/paper/2310.10767/) (2023)
- [Provable Guarantees for Neural Networks via Gradient Feature Learning](https://mlsanity.com/paper/2310.12408/) (2023)
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- [GenMix: Effective Data Augmentation with Generative Diffusion Model Image Editing](https://mlsanity.com/paper/2412.02366/) (2024)
- [Quantifying the Optimization and Generalization Advantages of Graph Neural Networks Over Multilayer Perceptrons](https://mlsanity.com/paper/2306.13926/) (2023)
- [Understanding Benign Overfitting in Gradient-Based Meta Learning](https://mlsanity.com/paper/2206.13482/) (2022)
- [Noisy Interpolation Learning with Shallow Univariate ReLU Networks](https://mlsanity.com/paper/2307.15396/) (2023)
- [Optimal Rate of Kernel Regression in Large Dimensions](https://mlsanity.com/paper/2309.04268/) (2023)

## Abstract (arXiv)

Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance. This phenomenon is referred to as "benign overfitting". Recently, there emerges a line of works studying "benign overfitting" from the theoretical perspective. However, they are limited to linear models or kernel/random feature models, and there is still a lack of theoretical understanding about when and how benign overfitting occurs in neural networks. In this paper, we study the benign overfitting phenomenon in training a two-layer convolutional neural network (CNN). We show that when the signal-to-noise ratio satisfies a certain condition, a two-layer CNN trained by gradient descent can achieve arbitrarily small training and test loss. On the other hand, when this condition does not hold, overfitting becomes harmful and the obtained CNN can only achieve a constant level test loss. These together demonstrate a sharp phase transition between benign overfitting and harmful overfitting, driven by the signal-to-noise ratio. To the best of our knowledge, this is the first work that precisely characterizes the conditions under which benign overfitting can occur in training convolutional neural networks.