# From Tempered to Benign Overfitting in ReLU Neural Networks

arXiv: https://arxiv.org/abs/2305.15141 · HTML: https://mlsanity.com/paper/2305.15141/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 59 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 30 of 44)

- [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)
- [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)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (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)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate](https://mlsanity.com/paper/1806.05161/) (2018)
- [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)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [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)
- [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)
- [Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon](https://mlsanity.com/paper/1812.11167/) (2018)
- [In Defense of Uniform Convergence: Generalization via derandomization with an application to interpolating predictors](https://mlsanity.com/paper/1912.04265/) (2019)
- [Distributional Generalization: A New Kind of Generalization](https://mlsanity.com/paper/2009.08092/) (2020)
- [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)
- [On Margin Maximization in Linear and ReLU Networks](https://mlsanity.com/paper/2110.02732/) (2021)
- [Theoretical Insights Into Multiclass Classification: A High-dimensional Asymptotic View](https://mlsanity.com/paper/2011.07729/) (2020)

## Built on by (most cited first, 16 of 16)

- [Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data](https://mlsanity.com/paper/2310.02541/) (2023)
- [Provable Guarantees for Neural Networks via Gradient Feature Learning](https://mlsanity.com/paper/2310.12408/) (2023)
- [Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes](https://mlsanity.com/paper/2406.06838/) (2024)
- [Noisy Interpolation Learning with Shallow Univariate ReLU Networks](https://mlsanity.com/paper/2307.15396/) (2023)
- [The Convex Landscape of Neural Networks: Characterizing Global Optima and Stationary Points via Lasso Models](https://mlsanity.com/paper/2312.12657/) (2023)
- [Overfitting Behaviour of Gaussian Kernel Ridgeless Regression: Varying Bandwidth or Dimensionality](https://mlsanity.com/paper/2409.03891/) (2024)
- [Benign overfitting in leaky ReLU networks with moderate input dimension](https://mlsanity.com/paper/2403.06903/) (2024)
- [Provable Tempered Overfitting of Minimal Nets and Typical Nets](https://mlsanity.com/paper/2410.19092/) (2024)
- [Neural Networks Generalize on Low Complexity Data](https://mlsanity.com/paper/2409.12446/) (2024)
- [Scanning Trojaned Models Using Out-of-Distribution Samples](https://mlsanity.com/paper/2501.17151/) (2025)
- [Generalization Below the Edge of Stability: The Role of Data Geometry](https://mlsanity.com/paper/2510.18120/) (2025)
- [Beyond Benign Overfitting in Nadaraya-Watson Interpolators](https://mlsanity.com/paper/2502.07480/) (2025)
- [Quantifying Overfitting along the Regularization Path for Two-Part-Code MDL in Supervised Classification](https://mlsanity.com/paper/2503.02110/) (2025)
- [A Library of Mirrors: Deep Neural Nets in Low Dimensions are Convex Lasso Models with Reflection Features](https://mlsanity.com/paper/2403.01046/) (2024)
- [Harmful Overfitting in Sobolev Spaces](https://mlsanity.com/paper/2602.00825/) (2026)
- [Neuron Block Dynamics for XOR Classification with Zero-Margin](https://mlsanity.com/paper/2602.00172/) (2026)

## Abstract (arXiv)

Overparameterized neural networks (NNs) are observed to generalize well even when trained to perfectly fit noisy data. This phenomenon motivated a large body of work on "benign overfitting", where interpolating predictors achieve near-optimal performance. Recently, it was conjectured and empirically observed that the behavior of NNs is often better described as "tempered overfitting", where the performance is non-optimal yet also non-trivial, and degrades as a function of the noise level. However, a theoretical justification of this claim for non-linear NNs has been lacking so far. In this work, we provide several results that aim at bridging these complementing views. We study a simple classification setting with 2-layer ReLU NNs, and prove that under various assumptions, the type of overfitting transitions from tempered in the extreme case of one-dimensional data, to benign in high dimensions. Thus, we show that the input dimension has a crucial role on the type of overfitting in this setting, which we also validate empirically for intermediate dimensions. Overall, our results shed light on the intricate connections between the dimension, sample size, architecture and training algorithm on the one hand, and the type of resulting overfitting on the other hand.