# Benign Overfitting in Deep Neural Networks under Lazy Training

arXiv: https://arxiv.org/abs/2305.19377 · HTML: https://mlsanity.com/paper/2305.19377/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 61 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 30 of 45)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [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)
- [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)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (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)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks](https://mlsanity.com/paper/2012.11654/) (2020)
- [How Much Over-parameterization Is Sufficient to Learn Deep ReLU Networks?](https://mlsanity.com/paper/1911.12360/) (2019)
- [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [Do Wider Neural Networks Really Help Adversarial Robustness?](https://mlsanity.com/paper/2010.01279/) (2020)
- [On the Similarity between the Laplace and Neural Tangent Kernels](https://mlsanity.com/paper/2007.01580/) (2020)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [Exploring Architectural Ingredients of Adversarially Robust Deep Neural Networks](https://mlsanity.com/paper/2110.03825/) (2021)
- [Benign Overfitting without Linearity: Neural Network Classifiers Trained by Gradient Descent for Noisy Linear Data](https://mlsanity.com/paper/2202.05928/) (2022)
- [Benign Overfitting of Constant-Stepsize SGD for Linear Regression](https://mlsanity.com/paper/2103.12692/) (2021)
- [Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1902.01384/) (2019)

## Built on by (most cited first, 10 of 10)

- [Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity](https://mlsanity.com/paper/2307.08286/) (2023)
- [On the Optimization and Generalization of Multi-head Attention](https://mlsanity.com/paper/2310.12680/) (2023)
- [The Surprising Harmfulness of Benign Overfitting for Adversarial Robustness](https://mlsanity.com/paper/2401.12236/) (2024)
- [Rote Learning Considered Useful: Generalizing over Memorized Data in LLMs](https://mlsanity.com/paper/2507.21914/) (2025)
- [Can overfitted deep neural networks in adversarial training generalize? -- An approximation viewpoint](https://mlsanity.com/paper/2401.13624/) (2024)
- [Neural Posterior Estimation with Autoregressive Tiling for Detecting Objects in Astronomical Images](https://mlsanity.com/paper/2510.03074/) (2025)
- [When Does ℓ2-Boosting Overfit Benignly? High-Dimensional Risk Asymptotics and the ℓ1 Implicit Bias](https://mlsanity.com/paper/2605.06314/) (2026)
- [Class-wise Activation Unravelling the Engima of Deep Double Descent](https://mlsanity.com/paper/2405.07679/) (2024)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Benign Overfitting under Learning Rate Conditions for α Sub-exponential Input](https://mlsanity.com/paper/2409.00733/) (2024)

## Abstract (arXiv)

This paper focuses on over-parameterized deep neural networks (DNNs) with ReLU activation functions and proves that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification while obtaining (nearly) zero-training error under the lazy training regime. For this purpose, we unify three interrelated concepts of overparameterization, benign overfitting, and the Lipschitz constant of DNNs. Our results indicate that interpolating with smoother functions leads to better generalization. Furthermore, we investigate the special case where interpolating smooth ground-truth functions is performed by DNNs under the Neural Tangent Kernel (NTK) regime for generalization. Our result demonstrates that the generalization error converges to a constant order that only depends on label noise and initialization noise, which theoretically verifies benign overfitting. Our analysis provides a tight lower bound on the normalized margin under non-smooth activation functions, as well as the minimum eigenvalue of NTK under high-dimensional settings, which has its own interest in learning theory.