# (S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability

arXiv: https://arxiv.org/abs/2302.08982 · HTML: https://mlsanity.com/paper/2302.08982/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 69 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 30 of 47)

- [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)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [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)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [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)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networks](https://mlsanity.com/paper/1907.04595/) (2019)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)
- [Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process](https://mlsanity.com/paper/1904.09080/) (2019)
- [Label Noise SGD Provably Prefers Flat Global Minimizers](https://mlsanity.com/paper/2106.06530/) (2021)
- [An Alternative View: When Does SGD Escape Local Minima?](https://mlsanity.com/paper/1802.06175/) (2018)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Revisiting Small Batch Training for Deep Neural Networks](https://mlsanity.com/paper/1804.07612/) (2018)
- [On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length](https://mlsanity.com/paper/1807.05031/) (2018)
- [Shape Matters: Understanding the Implicit Bias of the Noise Covariance](https://mlsanity.com/paper/2006.08680/) (2020)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent](https://mlsanity.com/paper/2102.09769/) (2021)

## Built on by (most cited first, 30 of 36)

- [The Surprising Agreement Between Convex Optimization Theory and Learning-Rate Scheduling for Large Model Training](https://mlsanity.com/paper/2501.18965/) (2025)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Implicit Regularization of Gradient Flow on One-Layer Softmax Attention](https://mlsanity.com/paper/2403.08699/) (2024)
- [Improving Generalization and Convergence by Enhancing Implicit Regularization](https://mlsanity.com/paper/2405.20763/) (2024)
- [Plastic Learning with Deep Fourier Features](https://mlsanity.com/paper/2410.20634/) (2024)
- [Mirror, Mirror of the Flow: How Does Regularization Shape Implicit Bias?](https://mlsanity.com/paper/2504.12883/) (2025)
- [Smoothing the Edges: Smooth Optimization for Sparse Regularization using Hadamard Overparametrization](https://mlsanity.com/paper/2307.03571/) (2023)
- [Implicit Bias of Mirror Flow on Separable Data](https://mlsanity.com/paper/2406.12763/) (2024)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)
- [Swing-by Dynamics in Concept Learning and Compositional Generalization](https://mlsanity.com/paper/2410.08309/) (2024)
- [Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks](https://mlsanity.com/paper/2307.07410/) (2023)
- [Never Saddle for Reparameterized Steepest Descent as Mirror Flow](https://mlsanity.com/paper/2603.02064/) (2026)
- [Towards Understanding Epoch-wise Double descent in Two-layer Linear Neural Networks](https://mlsanity.com/paper/2407.09845/) (2024)
- [Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective](https://mlsanity.com/paper/2601.06597/) (2026)
- [Edge Flow: A Tractable and Predictive Continuous-Time Model for Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2606.18080/) (2026)
- [The Effect of SGD Batch Size on Autoencoder Learning: Sparsity, Sharpness, and Feature Learning](https://mlsanity.com/paper/2308.03215/) (2023)
- [The Implicit Bias of Heterogeneity towards Invariance: A Study of Multi-Environment Matrix Sensing](https://mlsanity.com/paper/2403.01420/) (2024)
- [Optimization Insights into Deep Diagonal Linear Networks](https://mlsanity.com/paper/2412.16765/) (2024)
- [Variational Learning Finds Flatter Solutions at the Edge of Stability](https://mlsanity.com/paper/2506.12903/) (2025)
- [Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory](https://mlsanity.com/paper/2510.01930/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [Does SGD Seek Flatness or Sharpness? An Exactly Solvable Model](https://mlsanity.com/paper/2602.05065/) (2026)
- [On the implicit regularization of Langevin dynamics with projected noise](https://mlsanity.com/paper/2602.12257/) (2026)
- [Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension](https://mlsanity.com/paper/2602.12471/) (2026)
- [Large Spikes in Stochastic Gradient Descent: A Large-Deviations View](https://mlsanity.com/paper/2603.10079/) (2026)
- [Hard labels sampled from sparse targets mislead rotation invariant algorithms](https://mlsanity.com/paper/2603.20967/) (2026)
- [Gradient Flow Dynamics and Implicit Bias of Diagonal Linear Networks under Infinitesimal Initialization](https://mlsanity.com/paper/2607.12332/) (2026)
- [The Road Taken: The Role of Optimizers at the Edge of Stability](https://mlsanity.com/paper/2608.18415/) (2026)
- [Criteria and Bias of Parameterized Linear Regression under Edge of Stability Regime](https://mlsanity.com/paper/2412.08025/) (2024)
- [Over-Alignment vs Over-Fitting: The Role of Feature Learning Strength in Generalization](https://mlsanity.com/paper/2602.00827/) (2026)

## Abstract (arXiv)

In this paper, we investigate the impact of stochasticity and large stepsizes on the implicit regularisation of gradient descent (GD) and stochastic gradient descent (SGD) over diagonal linear networks. We prove the convergence of GD and SGD with macroscopic stepsizes in an overparametrised regression setting and characterise their solutions through an implicit regularisation problem. Our crisp characterisation leads to qualitative insights about the impact of stochasticity and stepsizes on the recovered solution. Specifically, we show that large stepsizes consistently benefit SGD for sparse regression problems, while they can hinder the recovery of sparse solutions for GD. These effects are magnified for stepsizes in a tight window just below the divergence threshold, in the "edge of stability" regime. Our findings are supported by experimental results.