# SGD with Large Step Sizes Learns Sparse Features

arXiv: https://arxiv.org/abs/2210.05337 · HTML: https://mlsanity.com/paper/2210.05337/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 62 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Builds on (most cited first, 30 of 49)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [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)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Sparsity in Deep Learning: Pruning and growth for efficient inference and training in neural networks](https://mlsanity.com/paper/2102.00554/) (2021)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Exploiting Linear Structure Within Convolutional Networks for Efficient Evaluation](https://mlsanity.com/paper/1404.0736/) (2014)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [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)
- [Understanding Batch Normalization](https://mlsanity.com/paper/1806.02375/) (2018)
- [Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process](https://mlsanity.com/paper/1904.09080/) (2019)
- [An Exponential Learning Rate Schedule for Deep Learning](https://mlsanity.com/paper/1910.07454/) (2019)
- [Label Noise SGD Provably Prefers Flat Global Minimizers](https://mlsanity.com/paper/2106.06530/) (2021)
- [Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations](https://mlsanity.com/paper/1811.01558/) (2018)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [What Happens after SGD Reaches Zero Loss? --A Mathematical Framework](https://mlsanity.com/paper/2110.06914/) (2021)
- [Shape Matters: Understanding the Implicit Bias of the Noise Covariance](https://mlsanity.com/paper/2006.08680/) (2020)

## Built on by (most cited first, 30 of 68)

- [The Lazy Neuron Phenomenon: On Emergence of Activation Sparsity in Transformers](https://mlsanity.com/paper/2210.06313/) (2022)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [Why Do We Need Weight Decay in Modern Deep Learning?](https://mlsanity.com/paper/2310.04415/) (2023)
- [Learning threshold neurons via the "edge of stability"](https://mlsanity.com/paper/2212.07469/) (2022)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [No Train No Gain: Revisiting Efficient Training Algorithms For Transformer-based Language Models](https://mlsanity.com/paper/2307.06440/) (2023)
- [Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks](https://mlsanity.com/paper/2306.04251/) (2023)
- [Sharpness-Aware Minimization Leads to Low-Rank Features](https://mlsanity.com/paper/2305.16292/) (2023)
- [(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability](https://mlsanity.com/paper/2302.08982/) (2023)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Spectral Evolution and Invariance in Linear-width Neural Networks](https://mlsanity.com/paper/2211.06506/) (2022)
- [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023)
- [Spurious Feature Diversification Improves Out-of-distribution Generalization](https://mlsanity.com/paper/2309.17230/) (2023)
- [Layer-wise Linear Mode Connectivity](https://mlsanity.com/paper/2307.06966/) (2023)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Fast TRAC: A Parameter-Free Optimizer for Lifelong Reinforcement Learning](https://mlsanity.com/paper/2405.16642/) (2024)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization](https://mlsanity.com/paper/2502.03435/) (2025)
- [Loss Spike in Training Neural Networks](https://mlsanity.com/paper/2305.12133/) (2023)
- [Towards a theory of model distillation](https://mlsanity.com/paper/2403.09053/) (2024)
- [Learning useful representations for shifting tasks and distributions](https://mlsanity.com/paper/2212.07346/) (2022)
- [Butterfly Effects of SGD Noise: Error Amplification in Behavior Cloning and Autoregression](https://mlsanity.com/paper/2310.11428/) (2023)
- [Deep Weight Factorization: Sparse Learning Through the Lens of Artificial Symmetries](https://mlsanity.com/paper/2502.02496/) (2025)
- [On the different regimes of Stochastic Gradient Descent](https://mlsanity.com/paper/2309.10688/) (2023)
- [Beyond Implicit Bias: The Insignificance of SGD Noise in Online Learning](https://mlsanity.com/paper/2306.08590/) (2023)
- [Sharpness-Aware Pretraining Mitigates Catastrophic Forgetting](https://mlsanity.com/paper/2605.02105/) (2026)
- [Learning on a Razor's Edge: Identifiability and Singularity of Polynomial Neural Networks](https://mlsanity.com/paper/2505.11846/) (2025)
- [Optimal scaling laws in learning hierarchical multi-index models](https://mlsanity.com/paper/2602.05846/) (2026)
- [Catapult Dynamics and Phase Transitions in Quadratic Nets](https://mlsanity.com/paper/2301.07737/) (2023)
- [When Do Neural Networks Learn World Models?](https://mlsanity.com/paper/2502.09297/) (2025)

## Abstract (arXiv)

We showcase important features of the dynamics of the Stochastic Gradient Descent (SGD) in the training of neural networks. We present empirical observations that commonly used large step sizes (i) lead the iterates to jump from one side of a valley to the other causing loss stabilization, and (ii) this stabilization induces a hidden stochastic dynamics orthogonal to the bouncing directions that biases it implicitly toward sparse predictors. Furthermore, we show empirically that the longer large step sizes keep SGD high in the loss landscape valleys, the better the implicit regularization can operate and find sparse representations. Notably, no explicit regularization is used so that the regularization effect comes solely from the SGD training dynamics influenced by the step size schedule. Therefore, these observations unveil how, through the step size schedules, both gradient and noise drive together the SGD dynamics through the loss landscape of neural networks. We justify these findings theoretically through the study of simple neural network models as well as qualitative arguments inspired from stochastic processes. Finally, this analysis allows us to shed a new light on some common practice and observed phenomena when training neural networks. The code of our experiments is available at https://github.com/tml-epfl/sgd-sparse-features.