# Directional Convergence, Benign Overfitting of Gradient Descent in leaky ReLU two-layer Neural Networks

arXiv: https://arxiv.org/abs/2505.16204 · HTML: https://mlsanity.com/paper/2505.16204/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 37 references that are in the map.
- First posted on arXiv 22 May 2025.

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

- [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)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [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)
- [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)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020)
- [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020)
- [Benign Overfitting without Linearity: Neural Network Classifiers Trained by Gradient Descent for Noisy Linear Data](https://mlsanity.com/paper/2202.05928/) (2022)
- [Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures](https://mlsanity.com/paper/2104.13628/) (2021)
- [Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data](https://mlsanity.com/paper/2210.07082/) (2022)
- [Benign Overfitting for Two-layer ReLU Convolutional Neural Networks](https://mlsanity.com/paper/2303.04145/) (2023)
- [On the proliferation of support vectors in high dimensions](https://mlsanity.com/paper/2009.10670/) (2020)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [Early Neuron Alignment in Two-layer ReLU Networks with Small Initialization](https://mlsanity.com/paper/2307.12851/) (2023)
- [Binary Classification of Gaussian Mixtures: Abundance of Support Vectors, Benign Overfitting and Regularization](https://mlsanity.com/paper/2011.09148/) (2020)
- [Towards Understanding Learning in Neural Networks with Linear Teachers](https://mlsanity.com/paper/2101.02533/) (2021)
- [Implicit Bias of Gradient Descent for Two-layer ReLU and Leaky ReLU Networks on Nearly-orthogonal Data](https://mlsanity.com/paper/2310.18935/) (2023)
- [Implicit Bias of Gradient Descent for Non-Homogeneous Deep Networks](https://mlsanity.com/paper/2502.16075/) (2025)
- [Benign overfitting in leaky ReLU networks with moderate input dimension](https://mlsanity.com/paper/2403.06903/) (2024)
- [Universality of Benign Overfitting in Binary Linear Classification](https://mlsanity.com/paper/2501.10538/) (2025)

## Abstract (arXiv)

In this paper, we provide sufficient conditions of benign overfitting of fixed width leaky ReLU two-layer neural network classifiers trained on mixture data via gradient descent. Our results are derived by establishing directional convergence of the network parameters and classification error bound of the convergent direction. Our classification error bound also lead to the discovery of a newly identified phase transition. Previously, directional convergence in (leaky) ReLU neural networks was established only for gradient flow. Due to the lack of directional convergence, previous results on benign overfitting were limited to those trained on nearly orthogonal data. All of our results hold on mixture data, which is a broader data setting than the nearly orthogonal data setting in prior work. We demonstrate our findings by showing that benign overfitting occurs with high probability in a much wider range of scenarios than previously known. Our results also allow us to characterize cases when benign overfitting provably fails even if directional convergence occurs. Our work thus provides a more complete picture of benign overfitting in leaky ReLU two-layer neural networks.