# Does SGD Seek Flatness or Sharpness? An Exactly Solvable Model

arXiv: https://arxiv.org/abs/2602.05065 · HTML: https://mlsanity.com/paper/2602.05065/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 59 references that are in the map.
- First posted on arXiv 4 Feb 2026.

## Builds on (most cited first, 30 of 50)

- [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)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [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 Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [The Break-Even Point on Optimization Trajectories of Deep Neural Networks](https://mlsanity.com/paper/2002.09572/) (2020)
- [Understanding Gradient Descent on Edge of Stability in Deep Learning](https://mlsanity.com/paper/2205.09745/) (2022)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [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)
- [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)
- [Adaptive Gradient Methods at the Edge of Stability](https://mlsanity.com/paper/2207.14484/) (2022)
- [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)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction](https://mlsanity.com/paper/2206.07085/) (2022)
- [Deep linear neural networks with arbitrary loss: All local minima are global](https://mlsanity.com/paper/1712.01473/) (2017)
- [The alignment property of SGD noise and how it helps select flat minima: A stability analysis](https://mlsanity.com/paper/2207.02628/) (2022)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stability](https://mlsanity.com/paper/2207.12678/) (2022)
- [Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers](https://mlsanity.com/paper/1910.05505/) (2019)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Width Provably Matters in Optimization for Deep Linear Neural Networks](https://mlsanity.com/paper/1901.08572/) (2019)
- [Learning threshold neurons via the "edge of stability"](https://mlsanity.com/paper/2212.07469/) (2022)

## Built on by (most cited first, 1 of 1)

- [Learning the identity: a case study of how SGD selects among functional decompositions](https://mlsanity.com/paper/2610.00615/) (2026)

## Abstract (arXiv)

A large body of theory and empirical work hypothesizes a connection between the flatness of a neural network's loss landscape during training and its performance. However, there have been conceptually opposite pieces of evidence regarding when SGD prefers flatter or sharper solutions during training. In this work, we partially but causally clarify the flatness-seeking behavior of SGD by identifying and exactly solving an analytically solvable model that exhibits both flattening and sharpening behavior during training. In this model, the SGD training has no textita priori preference for flatness, but only a preference for minimal gradient fluctuations. This leads to the insight that, at least within this model, it is data distribution that uniquely determines the sharpness at convergence, and that a flat minimum is preferred if and only if the noise in the labels is isotropic across all output dimensions. When the noise in the labels is anisotropic, the model instead prefers sharpness and can converge to an arbitrarily sharp solution, depending on the imbalance in the noise in the labels spectrum. We reproduce this key insight in controlled settings with different model architectures such as MLP, RNN, and transformers.