# The Rich and the Simple: On the Implicit Bias of Adam and SGD

arXiv: https://arxiv.org/abs/2505.24022 · HTML: https://mlsanity.com/paper/2505.24022/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 63 references that are in the map.
- First posted on arXiv 29 May 2025.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [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)
- [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)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Nuanced Metrics for Measuring Unintended Bias with Real Data for Text Classification](https://mlsanity.com/paper/1903.04561/) (2019)
- [Last Layer Re-Training is Sufficient for Robustness to Spurious Correlations](https://mlsanity.com/paper/2204.02937/) (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)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [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)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning](https://mlsanity.com/paper/2010.05627/) (2020)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [Convergence of Gradient Descent on Separable Data](https://mlsanity.com/paper/1803.01905/) (2018)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [Agree to Disagree: Diversity through Disagreement for Better Transferability](https://mlsanity.com/paper/2202.04414/) (2022)

## Built on by (most cited first, 13 of 13)

- [How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Data](https://mlsanity.com/paper/2510.22980/) (2025)
- [To Use or not to Use Muon: How Simplicity Bias in Optimizers Matters](https://mlsanity.com/paper/2603.00742/) (2026)
- [Why is Your Language Model a Poor Implicit Reward Model?](https://mlsanity.com/paper/2507.07981/) (2025)
- [How do simple rotations affect the implicit bias of Adam?](https://mlsanity.com/paper/2510.23804/) (2025)
- [OLion: Approaching the Hadamard Ideal by Intersecting Spectral and ℓ∞ Implicit Biases](https://mlsanity.com/paper/2602.01105/) (2026)
- [How to Set β1, β2 in Adam: An Online Learning Perspective](https://mlsanity.com/paper/2510.03478/) (2025)
- [Implicit Bias of Per-sample Adam on Separable Data: Departure from the Full-batch Regime](https://mlsanity.com/paper/2510.26303/) (2025)
- [When and Why SignSGD Outperforms SGD: A Theoretical Study Based on ℓ1-norm Lower Bounds](https://mlsanity.com/paper/2605.06615/) (2026)
- [Large Spikes in Stochastic Gradient Descent: A Large-Deviations View](https://mlsanity.com/paper/2603.10079/) (2026)
- [The Anatomy of Implicit Bias: Information Allocation in Neural Network Training](https://mlsanity.com/paper/2607.07156/) (2026)
- [Non-asymptotic implicit bias of logistic regression at early-stage gradient descent dynamics](https://mlsanity.com/paper/2608.04382/) (2026)
- [The Conflict Between Logic and Memory: Training Conditions for Optimizer-Dependent Rule Acquisition](https://mlsanity.com/paper/2610.00403/) (2026)
- [Optimizer Geometry Sets the Pace: Spectral Learning Dynamics in Matrix Factorization](https://mlsanity.com/paper/2610.04249/) (2026)

## Abstract (arXiv)

Adam is the de facto optimization algorithm for several deep learning applications, but an understanding of its implicit bias and how it differs from other algorithms, particularly standard first-order methods such as (stochastic) gradient descent (GD), remains limited. In practice, neural networks (NNs) trained with SGD are known to exhibit simplicity bias -- a tendency to find simple solutions. In contrast, we show that Adam is more resistant to such simplicity bias. First, we investigate the differences in the implicit biases of Adam and GD when training two-layer ReLU NNs on a binary classification task with Gaussian data. We find that GD exhibits a simplicity bias, resulting in a linear decision boundary with a suboptimal margin, whereas Adam leads to much richer and more diverse features, producing a nonlinear boundary that is closer to the Bayes' optimal predictor. This richer decision boundary also allows Adam to achieve higher test accuracy both in-distribution and under certain distribution shifts. We theoretically prove these results by analyzing the population gradients. Next, to corroborate our theoretical findings, we present extensive empirical results showing that this property of Adam leads to superior generalization across various datasets with spurious correlations where NNs trained with SGD are known to show simplicity bias and do not generalize well under certain distributional shifts.