# Bias in Motion: Theoretical Insights into the Dynamics of Bias in SGD Training

arXiv: https://arxiv.org/abs/2405.18296 · HTML: https://mlsanity.com/paper/2405.18296/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 47 references that are in the map.
- First posted on arXiv 28 May 2024.

## Builds on (most cited first, 28 of 28)

- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Just Train Twice: Improving Group Robustness without Training Group Information](https://mlsanity.com/paper/2107.09044/) (2021)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Fairness in Criminal Justice Risk Assessments: The State of the Art](https://mlsanity.com/paper/1703.09207/) (2017)
- [An Investigation of Why Overparameterization Exacerbates Spurious Correlations](https://mlsanity.com/paper/2005.04345/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [A Framework for Understanding Sources of Harm throughout the Machine Learning Life Cycle](https://mlsanity.com/paper/1901.10002/) (2019)
- [Understanding the Failure Modes of Out-of-Distribution Generalization](https://mlsanity.com/paper/2010.15775/) (2020)
- [Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup](https://mlsanity.com/paper/1906.08632/) (2019)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [On the Foundations of Shortcut Learning](https://mlsanity.com/paper/2310.16228/) (2023)
- [Neural networks trained with SGD learn distributions of increasing complexity](https://mlsanity.com/paper/2211.11567/) (2022)
- [Understanding training and generalization in deep learning by Fourier analysis](https://mlsanity.com/paper/1808.04295/) (2018)
- [Identifying Spurious Biases Early in Training through the Lens of Simplicity Bias](https://mlsanity.com/paper/2305.18761/) (2023)
- [From high-dimensional & mean-field dynamics to dimensionless ODEs: A unifying approach to SGD in two-layers networks](https://mlsanity.com/paper/2302.05882/) (2023)
- [On The Impact of Machine Learning Randomness on Group Fairness](https://mlsanity.com/paper/2307.04138/) (2023)
- [Procrustean Training for Imbalanced Deep Learning](https://mlsanity.com/paper/2104.01769/) (2021)
- [Asymptotic Bayes risk for Gaussian mixture in a semi-supervised setting](https://mlsanity.com/paper/1907.03792/) (2019)
- [Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering](https://mlsanity.com/paper/1610.02918/) (2016)
- [Simplicity Bias Leads to Amplified Performance Disparities](https://mlsanity.com/paper/2212.06641/) (2022)
- [Bias in Pruned Vision Models: In-Depth Analysis and Countermeasures](https://mlsanity.com/paper/2304.12622/) (2023)
- [Initial Guessing Bias: How Untrained Networks Favor Some Classes](https://mlsanity.com/paper/2306.00809/) (2023)

## Built on by (most cited first, 8 of 8)

- [Optimal Protocols for Continual Learning via Statistical Physics and Control Theory](https://mlsanity.com/paper/2409.18061/) (2024)
- [A theoretical perspective on mode collapse in variational inference](https://mlsanity.com/paper/2410.13300/) (2024)
- [Do We Always Need the Simplicity Bias? Looking for Optimal Inductive Biases in the Wild](https://mlsanity.com/paper/2503.10065/) (2025)
- [A Theory of Initialisation's Impact on Specialisation](https://mlsanity.com/paper/2503.02526/) (2025)
- [A statistical physics framework for optimal learning](https://mlsanity.com/paper/2507.07907/) (2025)
- [Reward Models Inherit Value Biases from Pretraining](https://mlsanity.com/paper/2601.20838/) (2026)
- [Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks](https://mlsanity.com/paper/2604.09412/) (2026)
- [Deciphering Shortcut Learning from an Evolutionary Game Theory Perspective](https://mlsanity.com/paper/2605.02658/) (2026)

## Abstract (arXiv)

Machine learning systems often acquire biases by leveraging undesired features in the data, impacting accuracy variably across different sub-populations. Current understanding of bias formation mostly focuses on the initial and final stages of learning, leaving a gap in knowledge regarding the transient dynamics. To address this gap, this paper explores the evolution of bias in a teacher-student setup modeling different data sub-populations with a Gaussian-mixture model. We provide an analytical description of the stochastic gradient descent dynamics of a linear classifier in this setting, which we prove to be exact in high dimension. Notably, our analysis reveals how different properties of sub-populations influence bias at different timescales, showing a shifting preference of the classifier during training. Applying our findings to fairness and robustness, we delineate how and when heterogeneous data and spurious features can generate and amplify bias. We empirically validate our results in more complex scenarios by training deeper networks on synthetic and real datasets, including CIFAR10, MNIST, and CelebA.