# Benign overfitting in ridge regression

arXiv: https://arxiv.org/abs/2009.14286 · HTML: https://mlsanity.com/paper/2009.14286/

## Key facts

- Cited by 153 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2020.

## Built on by (most cited first, 30 of 153)

- [Generalization Error Rates in Kernel Regression: The Crossover from the Noiseless to Noisy Regime](https://mlsanity.com/paper/2105.15004/) (2021)
- [How Many Pretraining Tasks Are Needed for In-Context Learning of Linear Regression?](https://mlsanity.com/paper/2310.08391/) (2023)
- [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)
- [A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning](https://mlsanity.com/paper/2109.02355/) (2021)
- [Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting](https://mlsanity.com/paper/2106.09276/) (2021)
- [Predicting Out-of-Distribution Error with the Projection Norm](https://mlsanity.com/paper/2202.05834/) (2022)
- [Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures](https://mlsanity.com/paper/2104.13628/) (2021)
- [Dual Operating Modes of In-Context Learning](https://mlsanity.com/paper/2402.18819/) (2024)
- [Benign Overfitting in Multiclass Classification: All Roads Lead to Interpolation](https://mlsanity.com/paper/2106.10865/) (2021)
- [Last Iterate Risk Bounds of SGD with Decaying Stepsize for Overparameterized Linear Regression](https://mlsanity.com/paper/2110.06198/) (2021)
- [Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting](https://mlsanity.com/paper/2207.06569/) (2022)
- [The Benefits of Implicit Regularization from SGD in Least Squares Problems](https://mlsanity.com/paper/2108.04552/) (2021)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties](https://mlsanity.com/paper/2205.07069/) (2022)
- [The Power and Limitation of Pretraining-Finetuning for Linear Regression under Covariate Shift](https://mlsanity.com/paper/2208.01857/) (2022)
- [Foolish Crowds Support Benign Overfitting](https://mlsanity.com/paper/2110.02914/) (2021)
- [The good, the bad and the ugly sides of data augmentation: An implicit spectral regularization perspective](https://mlsanity.com/paper/2210.05021/) (2022)
- [Fast Rates for Noisy Interpolation Require Rethinking the Effects of Inductive Bias](https://mlsanity.com/paper/2203.03597/) (2022)
- [High-dimensional Analysis of Knowledge Distillation: Weak-to-Strong Generalization and Scaling Laws](https://mlsanity.com/paper/2410.18837/) (2024)
- [Six Lectures on Linearized Neural Networks](https://mlsanity.com/paper/2308.13431/) (2023)
- [The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks](https://mlsanity.com/paper/2108.11489/) (2021)
- [More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory](https://mlsanity.com/paper/2311.14646/) (2023)
- [A Theoretical Analysis of Fine-tuning with Linear Teachers](https://mlsanity.com/paper/2107.01641/) (2021)
- [Optimistic Rates: A Unifying Theory for Interpolation Learning and Regularization in Linear Regression](https://mlsanity.com/paper/2112.04470/) (2021)
- [Tight bounds for minimum l1-norm interpolation of noisy data](https://mlsanity.com/paper/2111.05987/) (2021)
- [An elementary analysis of ridge regression with random design](https://mlsanity.com/paper/2203.08564/) (2022)
- [Harmless interpolation in regression and classification with structured features](https://mlsanity.com/paper/2111.05198/) (2021)
- [Parameters or Privacy: A Provable Tradeoff Between Overparameterization and Membership Inference](https://mlsanity.com/paper/2202.01243/) (2022)
- [How Transformers Utilize Multi-Head Attention in In-Context Learning? A Case Study on Sparse Linear Regression](https://mlsanity.com/paper/2408.04532/) (2024)

## Abstract (arXiv)

In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying overparameterized models. One of the central phenomena in this regime is the ability of the model to interpolate noisy data, but still have test error lower than the amount of noise in that data. arXiv:1906.11300 characterized for which covariance structure of the data such a phenomenon can happen in linear regression if one considers the interpolating solution with minimum ℓ2-norm and the data has independent components: they gave a sharp bound on the variance term and showed that it can be small if and only if the data covariance has high effective rank in a subspace of small co-dimension. We strengthen and complete their results by eliminating the independence assumption and providing sharp bounds for the bias term. Thus, our results apply in a much more general setting than those of arXiv:1906.11300, e.g., kernel regression, and not only characterize how the noise is damped but also which part of the true signal is learned. Moreover, we extend the result to the setting of ridge regression, which allows us to explain another interesting phenomenon: we give general sufficient conditions under which the optimal regularization is negative.