# Deep Linear Networks can Benignly Overfit when Shallow Ones Do arXiv: https://arxiv.org/abs/2209.09315 · HTML: https://mlsanity.com/paper/2209.09315/ ## Key facts - Cited by 8 papers in the map (data through 14 Oct 2026). - Builds on 61 of its 75 references that are in the map. - First posted on arXiv 19 Sep 2022. ## Builds on (most cited first, 30 of 61) - [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) - [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013) - [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) - [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014) - [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017) - [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020) - [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018) - [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018) - [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018) - [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021) - [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016) - [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018) - [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018) - [Benign overfitting in ridge regression](https://mlsanity.com/paper/2009.14286/) (2020) - [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020) - [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021) - [Harmless interpolation of noisy data in regression](https://mlsanity.com/paper/1903.09139/) (2019) - [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020) - [Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate](https://mlsanity.com/paper/1806.05161/) (2018) - [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) - [Does data interpolation contradict statistical optimality?](https://mlsanity.com/paper/1806.09471/) (2018) - [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022) - [A Model of Double Descent for High-dimensional Binary Linear Classification](https://mlsanity.com/paper/1911.05822/) (2019) - [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020) ## Built on by (most cited first, 8 of 8) - [Benign Overfitting for Two-layer ReLU Convolutional Neural Networks](https://mlsanity.com/paper/2303.04145/) (2023) - [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023) - [From Tempered to Benign Overfitting in ReLU Neural Networks](https://mlsanity.com/paper/2305.15141/) (2023) - [Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data](https://mlsanity.com/paper/2310.01975/) (2023) - [Precise Asymptotic Generalization for Multiclass Classification with Overparameterized Linear Models](https://mlsanity.com/paper/2306.13255/) (2023) - [Efficient Compression of Overparameterized Deep Models through Low-Dimensional Learning Dynamics](https://mlsanity.com/paper/2311.05061/) (2023) - [Towards Understanding Epoch-wise Double descent in Two-layer Linear Neural Networks](https://mlsanity.com/paper/2407.09845/) (2024) - [Understanding Private Learning From Feature Perspective](https://mlsanity.com/paper/2511.18006/) (2025) ## Abstract (arXiv) We bound the excess risk of interpolating deep linear networks trained using gradient flow. In a setting previously used to establish risk bounds for the minimum ℓ2-norm interpolant, we show that randomly initialized deep linear networks can closely approximate or even match known bounds for the minimum ℓ2-norm interpolant. Our analysis also reveals that interpolating deep linear models have exactly the same conditional variance as the minimum ℓ2-norm solution. Since the noise affects the excess risk only through the conditional variance, this implies that depth does not improve the algorithm's ability to "hide the noise". Our simulations verify that aspects of our bounds reflect typical behavior for simple data distributions. We also find that similar phenomena are seen in simulations with ReLU networks, although the situation there is more nuanced.