# Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction

arXiv: https://arxiv.org/abs/2206.07085 · HTML: https://mlsanity.com/paper/2206.07085/

## Key facts

- Cited by 86 papers in the map (data through 14 Oct 2026).
- Builds on 99 of its 129 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 30 of 99)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [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)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [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)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [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)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [How Does Batch Normalization Help Optimization?](https://mlsanity.com/paper/1805.11604/) (2018)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [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)

## Built on by (most cited first, 30 of 86)

- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Same Pre-training Loss, Better Downstream: Implicit Bias Matters for Language Models](https://mlsanity.com/paper/2210.14199/) (2022)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Transformers without Normalization](https://mlsanity.com/paper/2503.10622/) (2025)
- [How Does Sharpness-Aware Minimization Minimize Sharpness?](https://mlsanity.com/paper/2211.05729/) (2022)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Revisiting Weighted Aggregation in Federated Learning with Neural Networks](https://mlsanity.com/paper/2302.10911/) (2023)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Learning threshold neurons via the "edge of stability"](https://mlsanity.com/paper/2212.07469/) (2022)
- [Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing](https://mlsanity.com/paper/2301.11500/) (2023)
- [Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability](https://mlsanity.com/paper/2305.11788/) (2023)
- [Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalization](https://mlsanity.com/paper/2307.11007/) (2023)
- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency](https://mlsanity.com/paper/2402.15926/) (2024)
- [Normalization Layers Are All That Sharpness-Aware Minimization Needs](https://mlsanity.com/paper/2306.04226/) (2023)
- [Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory](https://mlsanity.com/paper/2307.04204/) (2023)
- [Good regularity creates large learning rate implicit biases: edge of stability, balancing, and catapult](https://mlsanity.com/paper/2310.17087/) (2023)
- [Sharpness-Aware Minimization Efficiently Selects Flatter Minima Late in Training](https://mlsanity.com/paper/2410.10373/) (2024)
- [Analyzing & Reducing the Need for Learning Rate Warmup in GPT Training](https://mlsanity.com/paper/2410.23922/) (2024)
- [A Multi-Power Law for Loss Curve Prediction Across Learning Rate Schedules](https://mlsanity.com/paper/2503.12811/) (2025)
- [Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond](https://mlsanity.com/paper/2305.13064/) (2023)
- [Explicit Eigenvalue Regularization Improves Sharpness-Aware Minimization](https://mlsanity.com/paper/2501.12666/) (2025)
- [The Marginal Value of Momentum for Small Learning Rate SGD](https://mlsanity.com/paper/2307.15196/) (2023)
- [From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression](https://mlsanity.com/paper/2310.01687/) (2023)
- [Large Stepsize Gradient Descent for Non-Homogeneous Two-Layer Networks: Margin Improvement and Fast Optimization](https://mlsanity.com/paper/2406.08654/) (2024)
- [The Geometry of Neural Nets' Parameter Spaces Under Reparametrization](https://mlsanity.com/paper/2302.07384/) (2023)
- [Loss Spike in Training Neural Networks](https://mlsanity.com/paper/2305.12133/) (2023)
- [Latent State Models of Training Dynamics](https://mlsanity.com/paper/2308.09543/) (2023)

## Abstract (arXiv)

Normalization layers (e.g., Batch Normalization, Layer Normalization) were introduced to help with optimization difficulties in very deep nets, but they clearly also help generalization, even in not-so-deep nets. Motivated by the long-held belief that flatter minima lead to better generalization, this paper gives mathematical analysis and supporting experiments suggesting that normalization (together with accompanying weight-decay) encourages GD to reduce the sharpness of loss surface. Here "sharpness" is carefully defined given that the loss is scale-invariant, a known consequence of normalization. Specifically, for a fairly broad class of neural nets with normalization, our theory explains how GD with a finite learning rate enters the so-called Edge of Stability (EoS) regime, and characterizes the trajectory of GD in this regime via a continuous sharpness-reduction flow.