# Benign overfitting in leaky ReLU networks with moderate input dimension

arXiv: https://arxiv.org/abs/2403.06903 · HTML: https://mlsanity.com/paper/2403.06903/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 39 references that are in the map.
- First posted on arXiv 11 Mar 2024.

## Builds on (most cited first, 30 of 31)

- [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)
- [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)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (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)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [On the Optimal Weighted ℓ2 Regularization in Overparameterized Linear Regression](https://mlsanity.com/paper/2006.05800/) (2020)
- [Classification vs regression in overparameterized regimes: Does the loss function matter?](https://mlsanity.com/paper/2005.08054/) (2020)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [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)
- [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)
- [Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting](https://mlsanity.com/paper/2106.09276/) (2021)
- [Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures](https://mlsanity.com/paper/2104.13628/) (2021)
- [Benign Overfitting in Multiclass Classification: All Roads Lead to Interpolation](https://mlsanity.com/paper/2106.10865/) (2021)
- [The Implicit Bias of Benign Overfitting](https://mlsanity.com/paper/2201.11489/) (2022)
- [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)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime](https://mlsanity.com/paper/1911.01544/) (2019)
- [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023)
- [Foolish Crowds Support Benign Overfitting](https://mlsanity.com/paper/2110.02914/) (2021)
- [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023)
- [Training shallow ReLU networks on noisy data using hinge loss: when do we overfit and is it benign?](https://mlsanity.com/paper/2306.09955/) (2023)

## Built on by (most cited first, 6 of 6)

- [Overfitting Behaviour of Gaussian Kernel Ridgeless Regression: Varying Bandwidth or Dimensionality](https://mlsanity.com/paper/2409.03891/) (2024)
- [Provable Tempered Overfitting of Minimal Nets and Typical Nets](https://mlsanity.com/paper/2410.19092/) (2024)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [Harmful Overfitting in Sobolev Spaces](https://mlsanity.com/paper/2602.00825/) (2026)
- [Directional Convergence, Benign Overfitting of Gradient Descent in leaky ReLU two-layer Neural Networks](https://mlsanity.com/paper/2505.16204/) (2025)
- [Double descent for least-squares interpolation on contaminated data: A simulation study](https://mlsanity.com/paper/2605.21494/) (2026)

## Abstract (arXiv)

The problem of benign overfitting asks whether it is possible for a model to perfectly fit noisy training data and still generalize well. We study benign overfitting in two-layer leaky ReLU networks trained with the hinge loss on a binary classification task. We consider input data that can be decomposed into the sum of a common signal and a random noise component, that lie on subspaces orthogonal to one another. We characterize conditions on the signal to noise ratio (SNR) of the model parameters giving rise to benign versus non-benign (or harmful) overfitting: in particular, if the SNR is high then benign overfitting occurs, conversely if the SNR is low then harmful overfitting occurs. We attribute both benign and non-benign overfitting to an approximate margin maximization property and show that leaky ReLU networks trained on hinge loss with gradient descent (GD) satisfy this property. In contrast to prior work we do not require the training data to be nearly orthogonal. Notably, for input dimension d and training sample size n, while results in prior work require d = Omega(n2 log n), here we require only d = Omegaleft(nright).