# The Implicit Bias of Adam on Separable Data

arXiv: https://arxiv.org/abs/2406.10650 · HTML: https://mlsanity.com/paper/2406.10650/

## Key facts

- Cited by 36 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 15 Jun 2024.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [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)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning](https://mlsanity.com/paper/2010.05627/) (2020)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization](https://mlsanity.com/paper/1808.05671/) (2018)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [The Implicit Regularization of Stochastic Gradient Flow for Least Squares](https://mlsanity.com/paper/2003.07802/) (2020)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [Toward Understanding Why Adam Converges Faster Than SGD for Transformers](https://mlsanity.com/paper/2306.00204/) (2023)
- [Lion Secretly Solves Constrained Optimization: As Lyapunov Predicts](https://mlsanity.com/paper/2310.05898/) (2023)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Understanding the Generalization of Adam in Learning Neural Networks with Proper Regularization](https://mlsanity.com/paper/2108.11371/) (2021)
- [The Implicit Bias for Adaptive Optimization Algorithms on Homogeneous Neural Networks](https://mlsanity.com/paper/2012.06244/) (2020)

## Built on by (most cited first, 30 of 36)

- [Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data](https://mlsanity.com/paper/2502.04664/) (2025)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Data](https://mlsanity.com/paper/2510.22980/) (2025)
- [Flavors of Margin: Implicit Bias of Steepest Descent in Homogeneous Neural Networks](https://mlsanity.com/paper/2410.22069/) (2024)
- [The Rich and the Simple: On the Implicit Bias of Adam and SGD](https://mlsanity.com/paper/2505.24022/) (2025)
- [The Implicit Bias of Adam and Muon on Smooth Homogeneous Neural Networks](https://mlsanity.com/paper/2602.16340/) (2026)
- [Can Muon Fine-tune Adam-Pretrained Models?](https://mlsanity.com/paper/2605.10468/) (2026)
- [Why is Your Language Model a Poor Implicit Reward Model?](https://mlsanity.com/paper/2507.07981/) (2025)
- [A Tale of Two Geometries: Adaptive Optimizers and Non-Euclidean Descent](https://mlsanity.com/paper/2511.20584/) (2025)
- [Continuous-Time Analysis of Adaptive Optimization and Normalization](https://mlsanity.com/paper/2411.05746/) (2024)
- [Towards Understanding Generalization in DP-GD: A Case Study in Training Two-Layer CNNs](https://mlsanity.com/paper/2511.22270/) (2025)
- [Never Saddle for Reparameterized Steepest Descent as Mirror Flow](https://mlsanity.com/paper/2603.02064/) (2026)
- [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)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [Towards Simple and Provable Parameter-Free Adaptive Gradient Methods](https://mlsanity.com/paper/2412.19444/) (2024)
- [Implicit Bias of Per-sample Adam on Separable Data: Departure from the Full-batch Regime](https://mlsanity.com/paper/2510.26303/) (2025)
- [Diagonalizing the Softmax: Hadamard Initialization for Tractable Cross-Entropy Dynamics](https://mlsanity.com/paper/2512.04006/) (2025)
- [Effectiveness of Distributed Gradient Descent with Local Steps for Overparameterized Models](https://mlsanity.com/paper/2412.07971/) (2024)
- [First-Passage Prediction of Grokking Delay: ACalibrated Law under AdamW with Causal Validation](https://mlsanity.com/paper/2605.18845/) (2026)
- [Gradient Descent Robustly Learns the Intrinsic Dimension of Data in Training Convolutional Neural Networks](https://mlsanity.com/paper/2504.08628/) (2025)
- [Heavy-Ball Momentum Method in Continuous Time and Discretization Error Analysis](https://mlsanity.com/paper/2506.14806/) (2025)
- [When Are Learning Biases Equivalent? A Unifying Framework for Fairness, Robustness, and Distribution Shift](https://mlsanity.com/paper/2511.07485/) (2025)
- [Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime](https://mlsanity.com/paper/2603.10485/) (2026)
- [Towards The Implicit Bias on Multiclass Separable Data Under Norm Constraints](https://mlsanity.com/paper/2603.22824/) (2026)
- [Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach](https://mlsanity.com/paper/2605.19392/) (2026)
- [HORST: Composing Optimizer Geometries for Sparse Transformer Training](https://mlsanity.com/paper/2605.21104/) (2026)
- [The Implicit Bias of Depth: From Neural Collapse to Softmax Codes](https://mlsanity.com/paper/2605.23087/) (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)

## Abstract (arXiv)

Adam has become one of the most favored optimizers in deep learning problems. Despite its success in practice, numerous mysteries persist regarding its theoretical understanding. In this paper, we study the implicit bias of Adam in linear logistic regression. Specifically, we show that when the training data are linearly separable, Adam converges towards a linear classifier that achieves the maximum ℓ_∞-margin. Notably, for a general class of diminishing learning rates, this convergence occurs within polynomial time. Our result shed light on the difference between Adam and (stochastic) gradient descent from a theoretical perspective.