# Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures

arXiv: https://arxiv.org/abs/2104.13628 · HTML: https://mlsanity.com/paper/2104.13628/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 38 references that are in the map.
- First posted on arXiv 28 Apr 2021.

## Builds on (most cited first, 20 of 20)

- [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)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [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)
- [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)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020)
- [A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent](https://mlsanity.com/paper/2006.05013/) (2020)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [On the proliferation of support vectors in high dimensions](https://mlsanity.com/paper/2009.10670/) (2020)

## Built on by (most cited first, 30 of 49)

- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [Benign Overfitting without Linearity: Neural Network Classifiers Trained by Gradient Descent for Noisy Linear Data](https://mlsanity.com/paper/2202.05928/) (2022)
- [A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning](https://mlsanity.com/paper/2109.02355/) (2021)
- [Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions](https://mlsanity.com/paper/2106.03791/) (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)
- [Towards an Understanding of Benign Overfitting in Neural Networks](https://mlsanity.com/paper/2106.03212/) (2021)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [The good, the bad and the ugly sides of data augmentation: An implicit spectral regularization perspective](https://mlsanity.com/paper/2210.05021/) (2022)
- [Is Importance Weighting Incompatible with Interpolating Classifiers?](https://mlsanity.com/paper/2112.12986/) (2021)
- [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023)
- [Harmless interpolation in regression and classification with structured features](https://mlsanity.com/paper/2111.05198/) (2021)
- [Implicit Optimization Bias of Next-Token Prediction in Linear Models](https://mlsanity.com/paper/2402.18551/) (2024)
- [Benign Overfitting in Adversarially Robust Linear Classification](https://mlsanity.com/paper/2112.15250/) (2021)
- [Noisy Interpolation Learning with Shallow Univariate ReLU Networks](https://mlsanity.com/paper/2307.15396/) (2023)
- [Multiple Descent in the Multiple Random Feature Model](https://mlsanity.com/paper/2208.09897/) (2022)
- [Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data](https://mlsanity.com/paper/2310.01975/) (2023)
- [Benefits of Early Stopping in Gradient Descent for Overparameterized Logistic Regression](https://mlsanity.com/paper/2502.13283/) (2025)
- [Support vector machines and linear regression coincide with very high-dimensional features](https://mlsanity.com/paper/2105.14084/) (2021)
- [Deep Linear Networks can Benignly Overfit when Shallow Ones Do](https://mlsanity.com/paper/2209.09315/) (2022)
- [Per-Example Gradient Regularization Improves Learning Signals from Noisy Data](https://mlsanity.com/paper/2303.17940/) (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)
- [Overfitting Behaviour of Gaussian Kernel Ridgeless Regression: Varying Bandwidth or Dimensionality](https://mlsanity.com/paper/2409.03891/) (2024)
- [Minimax Supervised Clustering in the Anisotropic Gaussian Mixture Model: A new take on Robust Interpolation](https://mlsanity.com/paper/2111.07041/) (2021)
- [Max-Margin Works while Large Margin Fails: Generalization without Uniform Convergence](https://mlsanity.com/paper/2206.07892/) (2022)
- [Malign Overfitting: Interpolation Can Provably Preclude Invariance](https://mlsanity.com/paper/2211.15724/) (2022)
- [Benign overfitting in leaky ReLU networks with moderate input dimension](https://mlsanity.com/paper/2403.06903/) (2024)
- [When and How Unlabeled Data Provably Improve In-Context Learning](https://mlsanity.com/paper/2506.15329/) (2025)
- [Benign Overfitting in Classification: Provably Counter Label Noise with Larger Models](https://mlsanity.com/paper/2206.00501/) (2022)

## Abstract (arXiv)

Modern machine learning systems such as deep neural networks are often highly over-parameterized so that they can fit the noisy training data exactly, yet they can still achieve small test errors in practice. In this paper, we study this "benign overfitting" phenomenon of the maximum margin classifier for linear classification problems. Specifically, we consider data generated from sub-Gaussian mixtures, and provide a tight risk bound for the maximum margin linear classifier in the over-parameterized setting. Our results precisely characterize the condition under which benign overfitting can occur in linear classification problems, and improve on previous work. They also have direct implications for over-parameterized logistic regression.