# On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions

arXiv: https://arxiv.org/abs/2402.03982 · HTML: https://mlsanity.com/paper/2402.03982/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 65 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 30 of 41)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018)
- [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)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization](https://mlsanity.com/paper/1808.05671/) (2018)
- [Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks](https://mlsanity.com/paper/1806.06763/) (2018)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [Global convergence of the Heavy-ball method for convex optimization](https://mlsanity.com/paper/1412.7457/) (2014)
- [A High Probability Analysis of Adaptive SGD with Momentum](https://mlsanity.com/paper/2007.14294/) (2020)
- [The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance](https://mlsanity.com/paper/2202.05791/) (2022)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [A Unified Analysis of Stochastic Momentum Methods for Deep Learning](https://mlsanity.com/paper/1808.10396/) (2018)
- [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)
- [High Probability Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2302.14843/) (2023)
- [Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD](https://mlsanity.com/paper/2302.06570/) (2023)

## Built on by (most cited first, 21 of 21)

- [LDAdam: Adaptive Optimization from Low-Dimensional Gradient Statistics](https://mlsanity.com/paper/2410.16103/) (2024)
- [The Implicit Bias of Adam on Separable Data](https://mlsanity.com/paper/2406.10650/) (2024)
- [AdaGrad under Anisotropic Smoothness](https://mlsanity.com/paper/2406.15244/) (2024)
- [Revisiting Convergence of AdaGrad with Relaxed Assumptions](https://mlsanity.com/paper/2402.13794/) (2024)
- [On the O(frac√dK1/4) Convergence Rate of AdamW Measured by ℓ1 Norm](https://mlsanity.com/paper/2505.11840/) (2025)
- [Continuous-Time Analysis of Adaptive Optimization and Normalization](https://mlsanity.com/paper/2411.05746/) (2024)
- [Simple Convergence Proof of Adam From a Sign-like Descent Perspective](https://mlsanity.com/paper/2507.05966/) (2025)
- [Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? A Case Study of AdaGrad](https://mlsanity.com/paper/2605.18694/) (2026)
- [Implicit Bias of Per-sample Adam on Separable Data: Departure from the Full-batch Regime](https://mlsanity.com/paper/2510.26303/) (2025)
- [MGUP: A Momentum-Gradient Alignment Update Policy for Stochastic Optimization](https://mlsanity.com/paper/2606.17526/) (2026)
- [Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds](https://mlsanity.com/paper/2505.15013/) (2025)
- [SoftSignSGD(S3): An Enhanced Optimizer for Practical DNN Training and Loss Spikes Minimization Beyond Adam](https://mlsanity.com/paper/2507.06464/) (2025)
- [Convergence Rate Analysis of the AdamW-style Shampoo: Unifying One-Sided and Two-Sided Preconditioning](https://mlsanity.com/paper/2601.07326/) (2026)
- [Adapt or Forget: Provable Tradeoffs Between Adam and SGD in Nonstationary Optimization](https://mlsanity.com/paper/2605.04269/) (2026)
- [Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?](https://mlsanity.com/paper/2606.23676/) (2026)
- [Analysis of Adam Algorithms for Stochastic Dynamic Systems](https://mlsanity.com/paper/2606.28879/) (2026)
- [Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach](https://mlsanity.com/paper/2605.19392/) (2026)
- [Convergence rates for the RMSprop optimizer with full control of the hyperparameters](https://mlsanity.com/paper/2608.30382/) (2026)
- [Why Clipping Matters in AdaGrad? Toward a High-Probability Theory under Generalized Smoothness](https://mlsanity.com/paper/2609.30276/) (2026)
- [Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds](https://mlsanity.com/paper/2609.32729/) (2026)
- [Adam under Generalized Smoothness with Second-Moment-Type Stochastic Gradients](https://mlsanity.com/paper/2609.37787/) (2026)

## Abstract (arXiv)

The Adaptive Momentum Estimation (Adam) algorithm is highly effective in training various deep learning tasks. Despite this, there's limited theoretical understanding for Adam, especially when focusing on its vanilla form in non-convex smooth scenarios with potential unbounded gradients and affine variance noise. In this paper, we study vanilla Adam under these challenging conditions. We introduce a comprehensive noise model which governs affine variance noise, bounded noise and sub-Gaussian noise. We show that Adam can find a stationary point with a O(poly(log T)/√T) rate in high probability under this general noise model where T denotes total number iterations, matching the lower rate of stochastic first-order algorithms up to logarithm factors. More importantly, we reveal that Adam is free of tuning step-sizes with any problem-parameters, yielding a better adaptation property than the Stochastic Gradient Descent under the same conditions. We also provide a probabilistic convergence result for Adam under a generalized smooth condition which allows unbounded smoothness parameters and has been illustrated empirically to more accurately capture the smooth property of many practical objective functions.