# Simplicity bias and optimization threshold in two-layer ReLU networks

arXiv: https://arxiv.org/abs/2410.02348 · HTML: https://mlsanity.com/paper/2410.02348/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 70 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 30 of 56)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [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)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)

## Built on by (most cited first, 11 of 11)

- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures](https://mlsanity.com/paper/2512.20607/) (2025)
- [A Theoretical Framework for Grokking: Interpolation followed by Riemannian Norm Minimisation](https://mlsanity.com/paper/2505.20172/) (2025)
- [Convergence of Shallow ReLU Networks on Weakly Interacting Data](https://mlsanity.com/paper/2502.16977/) (2025)
- [Favorability of Loss Landscape with Weight Decay Requires Both Large Overparametrization and Initialization](https://mlsanity.com/paper/2505.22578/) (2025)
- [Large Spikes in Stochastic Gradient Descent: A Large-Deviations View](https://mlsanity.com/paper/2603.10079/) (2026)
- [When Both Layers Learn: Training Dynamics of Representing Linear Models via ReLU Networks](https://mlsanity.com/paper/2606.04476/) (2026)
- [Generalization in Nonlinear Least Squares via Learned Feature Geometry](https://mlsanity.com/paper/2606.08799/) (2026)
- [Noise Stability of Transformer Models](https://mlsanity.com/paper/2602.08287/) (2026)
- [Do Deep Networks Forget Initialization? A Forgetting-Time View of Practical Inductive Bias](https://mlsanity.com/paper/2605.29152/) (2026)
- [Teacher Geometry Shapes Learnability in Teacher-Student Networks](https://mlsanity.com/paper/2609.09595/) (2026)

## Abstract (arXiv)

Understanding generalization of overparametrized neural networks remains a fundamental challenge in machine learning. Most of the literature mostly studies generalization from an interpolation point of view, taking convergence of parameters towards a global minimum of the training loss for granted. While overparametrized architectures indeed interpolated the data for typical classification tasks, this interpolation paradigm does not seem valid anymore for more complex tasks such as in-context learning or diffusion. Instead for such tasks, it has been empirically observed that the trained models goes from global minima to spurious local minima of the training loss as the number of training samples becomes larger than some level we call optimization threshold. While the former yields a poor generalization to the true population loss, the latter was observed to actually correspond to the minimiser of this true loss. This paper explores theoretically this phenomenon in the context of two-layer ReLU networks. We demonstrate that, despite overparametrization, networks often converge toward simpler solutions rather than interpolating the training data, which can lead to a drastic improvement on the test loss with respect to interpolating solutions. Our analysis relies on the so called early alignment phase, during which neurons align towards specific directions. This directional alignment, which occurs in the early stage of training, leads to a simplicity bias, wherein the network approximates the ground truth model without converging to the global minimum of the training loss. Our results suggest that this bias, resulting in an optimization threshold from which interpolation is not reached anymore, is beneficial and enhances the generalization of trained models.