# Simplicity Bias in 1-Hidden Layer Neural Networks

arXiv: https://arxiv.org/abs/2302.00457 · HTML: https://mlsanity.com/paper/2302.00457/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 68 references that are in the map.
- First posted on arXiv 1 Feb 2023.

## Builds on (most cited first, 30 of 48)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (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)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [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)
- [Just Train Twice: Improving Group Robustness without Training Group Information](https://mlsanity.com/paper/2107.09044/) (2021)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [ViM: Out-Of-Distribution with Virtual-logit Matching](https://mlsanity.com/paper/2203.10807/) (2022)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [An Investigation of Why Overparameterization Exacerbates Spurious Correlations](https://mlsanity.com/paper/2005.04345/) (2020)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Gradient Starvation: A Learning Proclivity in Neural Networks](https://mlsanity.com/paper/2011.09468/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradients](https://mlsanity.com/paper/1711.09404/) (2017)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [Intriguing Properties of Contrastive Losses](https://mlsanity.com/paper/2011.02803/) (2020)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)

## Built on by (most cited first, 26 of 26)

- [Feature emergence via margin maximization: case studies in algebraic tasks](https://mlsanity.com/paper/2311.07568/) (2023)
- [The Law of Parsimony in Gradient Descent for Learning Deep Linear Networks](https://mlsanity.com/paper/2306.01154/) (2023)
- [Transformers Learn Low Sensitivity Functions: Investigations and Implications](https://mlsanity.com/paper/2403.06925/) (2024)
- [The Rich and the Simple: On the Implicit Bias of Adam and SGD](https://mlsanity.com/paper/2505.24022/) (2025)
- [Complexity Matters: Dynamics of Feature Learning in the Presence of Spurious Correlations](https://mlsanity.com/paper/2403.03375/) (2024)
- [Project and Probe: Sample-Efficient Domain Adaptation by Interpolating Orthogonal Features](https://mlsanity.com/paper/2302.05441/) (2023)
- [Sharpness-Aware Minimization Enhances Feature Quality via Balanced Learning](https://mlsanity.com/paper/2405.20439/) (2024)
- [Spurious Correlations in High Dimensional Regression: The Roles of Regularization, Simplicity Bias and Over-Parameterization](https://mlsanity.com/paper/2502.01347/) (2025)
- [Mitigating Simplicity Bias in Deep Learning for Improved OOD Generalization and Robustness](https://mlsanity.com/paper/2310.06161/) (2023)
- [Severing Spurious Correlations with Data Pruning](https://mlsanity.com/paper/2503.18258/) (2025)
- [Spurious Correlation Learning in Preference Optimization: Mechanisms, Consequences, and Mitigation via Tie Training](https://mlsanity.com/paper/2605.11134/) (2026)
- [Understanding the dynamics of the frequency bias in neural networks](https://mlsanity.com/paper/2405.14957/) (2024)
- [Stagewise Reinforcement Learning and the Geometry of the Regret Landscape](https://mlsanity.com/paper/2601.07524/) (2026)
- [How do simple rotations affect the implicit bias of Adam?](https://mlsanity.com/paper/2510.23804/) (2025)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [A Unified Stability Analysis of SAM vs SGD: Role of Data Coherence and Emergence of Simplicity Bias](https://mlsanity.com/paper/2511.17378/) (2025)
- [Revealing Model Biases: Assessing Deep Neural Networks via Recovered Sample Analysis](https://mlsanity.com/paper/2306.06414/) (2023)
- [Hierarchical Simplicity Bias of Neural Networks](https://mlsanity.com/paper/2311.02622/) (2023)
- [Sequencing the Neurome: Towards Scalable Exact Parameter Reconstruction of Black-Box Neural Networks](https://mlsanity.com/paper/2409.19138/) (2024)
- [Convex Distillation: Efficient Compression of Deep Networks via Convex Optimization](https://mlsanity.com/paper/2410.06567/) (2024)
- [Impact of Label Noise on Learning Complex Features](https://mlsanity.com/paper/2411.04569/) (2024)
- [Divide and Conquer Self-Supervised Learning for High-Content Imaging](https://mlsanity.com/paper/2503.07444/) (2025)
- [A Modern Look at Simplicity Bias in Image Classification Tasks](https://mlsanity.com/paper/2509.12265/) (2025)
- [Noise Stability of Transformer Models](https://mlsanity.com/paper/2602.08287/) (2026)
- [Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion](https://mlsanity.com/paper/2605.29885/) (2026)
- [Whitening Improves Robustness to Spurious Correlations in Linear Probes](https://mlsanity.com/paper/2609.39177/) (2026)

## Abstract (arXiv)

Recent works have demonstrated that neural networks exhibit extreme simplicity bias(SB). That is, they learn only the simplest features to solve a task at hand, even in the presence of other, more robust but more complex features. Due to the lack of a general and rigorous definition of features, these works showcase SB on semi-synthetic datasets such as Color-MNIST, MNIST-CIFAR where defining features is relatively easier. In this work, we rigorously define as well as thoroughly establish SB for one hidden layer neural networks. More concretely, (i) we define SB as the network essentially being a function of a low dimensional projection of the inputs (ii) theoretically, we show that when the data is linearly separable, the network primarily depends on only the linearly separable (1-dimensional) subspace even in the presence of an arbitrarily large number of other, more complex features which could have led to a significantly more robust classifier, (iii) empirically, we show that models trained on real datasets such as Imagenette and Waterbirds-Landbirds indeed depend on a low dimensional projection of the inputs, thereby demonstrating SB on these datasets, iv) finally, we present a natural ensemble approach that encourages diversity in models by training successive models on features not used by earlier models, and demonstrate that it yields models that are significantly more robust to Gaussian noise.