# Simplicity Bias of Two-Layer Networks beyond Linearly Separable Data

arXiv: https://arxiv.org/abs/2405.17299 · HTML: https://mlsanity.com/paper/2405.17299/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 57 references that are in the map.
- First posted on arXiv 27 May 2024.

## Builds on (most cited first, 30 of 47)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [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)
- [Right for the Wrong Reasons: Diagnosing Syntactic Heuristics in Natural Language Inference](https://mlsanity.com/paper/1902.01007/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (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)
- [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)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel](https://mlsanity.com/paper/1810.05369/) (2018)
- [What shapes feature representations? Exploring datasets, architectures, and training](https://mlsanity.com/paper/2006.12433/) (2020)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Distilling Model Failures as Directions in Latent Space](https://mlsanity.com/paper/2206.14754/) (2022)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Phase diagram for two-layer ReLU neural networks at infinite-width limit](https://mlsanity.com/paper/2007.07497/) (2020)
- [Neural networks trained with SGD learn distributions of increasing complexity](https://mlsanity.com/paper/2211.11567/) (2022)
- [Theory of the Frequency Principle for General Deep Neural Networks](https://mlsanity.com/paper/1906.09235/) (2019)

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

- [Saddle-to-Saddle Dynamics Explains A Simplicity Bias Across Neural Network Architectures](https://mlsanity.com/paper/2512.20607/) (2025)
- [Do We Always Need the Simplicity Bias? Looking for Optimal Inductive Biases in the Wild](https://mlsanity.com/paper/2503.10065/) (2025)
- [Simplicity bias and optimization threshold in two-layer ReLU networks](https://mlsanity.com/paper/2410.02348/) (2024)
- [ALTA: Compiler-Based Analysis of Transformers](https://mlsanity.com/paper/2410.18077/) (2024)
- [Bridging Kolmogorov Complexity and Deep Learning: Asymptotically Optimal Description Length Objectives for Transformers](https://mlsanity.com/paper/2509.22445/) (2025)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [Review of Hallucination Understanding in Large Language and Vision Models](https://mlsanity.com/paper/2510.00034/) (2025)
- [Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data](https://mlsanity.com/paper/2510.21078/) (2025)
- [On Measuring Localization of Shortcuts in Deep Networks](https://mlsanity.com/paper/2510.26560/) (2025)
- [A Critical Review of Predominant Bias in Neural Networks](https://mlsanity.com/paper/2502.11031/) (2025)
- [Noise Stability of Transformer Models](https://mlsanity.com/paper/2602.08287/) (2026)

## Abstract (arXiv)

Simplicity bias, the propensity of deep models to over-rely on simple features, has been identified as a potential reason for limited out-of-distribution generalization of neural networks (Shah et al., 2020). Despite the important implications, this phenomenon has been theoretically confirmed and characterized only under strong dataset assumptions, such as linear separability (Lyu et al., 2021). In this work, we characterize simplicity bias for general datasets in the context of two-layer neural networks initialized with small weights and trained with gradient flow. Specifically, we prove that in the early training phases, network features cluster around a few directions that do not depend on the size of the hidden layer. Furthermore, for datasets with an XOR-like pattern, we precisely identify the learned features and demonstrate that simplicity bias intensifies during later training stages. These results indicate that features learned in the middle stages of training may be more useful for OOD transfer. We support this hypothesis with experiments on image data.