# Adam with model exponential moving average is effective for nonconvex optimization

arXiv: https://arxiv.org/abs/2405.18199 · HTML: https://mlsanity.com/paper/2405.18199/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 62 references that are in the map.
- First posted on arXiv 28 May 2024.

## Builds on (most cited first, 30 of 45)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Scaling up GANs for Text-to-Image Synthesis](https://mlsanity.com/paper/2303.05511/) (2023)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [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)
- [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)
- [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [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)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [The Unusual Effectiveness of Averaging in GAN Training](https://mlsanity.com/paper/1806.04498/) (2018)
- [Toward Understanding Why Adam Converges Faster Than SGD for Transformers](https://mlsanity.com/paper/2306.00204/) (2023)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Understanding AdamW through Proximal Methods and Scale-Freeness](https://mlsanity.com/paper/2202.00089/) (2022)

## Built on by (most cited first, 23 of 23)

- [Online Convex Optimization with Heavy Tails: Old Algorithms, New Regrets, and Applications](https://mlsanity.com/paper/2508.07473/) (2025)
- [General framework for online-to-nonconvex conversion: Schedule-free SGD is also effective for nonconvex optimization](https://mlsanity.com/paper/2411.07061/) (2024)
- [Adaptive Matrix Online Learning through Smoothing with Guarantees for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2602.08232/) (2026)
- [Through the River: Understanding the Benefit of Schedule-Free Methods for Language Model Training](https://mlsanity.com/paper/2507.09846/) (2025)
- [On the O(frac√dK1/4) Convergence Rate of AdamW Measured by ℓ1 Norm](https://mlsanity.com/paper/2505.11840/) (2025)
- [StoSignSGD: Unbiased Structural Stochasticity Fixes SignSGD for Training Large Language Models](https://mlsanity.com/paper/2604.15416/) (2026)
- [Muon with Finite Newton-Schulz: The Smoothing Benefit in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2608.26288/) (2026)
- [Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer](https://mlsanity.com/paper/2602.08372/) (2026)
- [Improving Online-to-Nonconvex Conversion for Smooth Optimization via Double Optimism](https://mlsanity.com/paper/2510.03167/) (2025)
- [Towards Simple and Provable Parameter-Free Adaptive Gradient Methods](https://mlsanity.com/paper/2412.19444/) (2024)
- [Elucidating Subspace Perturbation in Zeroth-Order Optimization: Theory and Practice at Scale](https://mlsanity.com/paper/2501.19099/) (2025)
- [PANDORA: Diffusion Policy Learning for Dexterous Robotic Piano Playing](https://mlsanity.com/paper/2503.14545/) (2025)
- [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)
- [The Convergence Behavior of Adam under Heavy-Tailed Noise](https://mlsanity.com/paper/2607.27383/) (2026)
- [PADAM: Parallel averaged Adam reduces the error for stochastic optimization in scientific machine learning](https://mlsanity.com/paper/2505.22085/) (2025)
- [Central limit theorem for the averaged Adam optimizer](https://mlsanity.com/paper/2606.21433/) (2026)
- [Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?](https://mlsanity.com/paper/2606.23676/) (2026)
- [Stochastic Non-Smooth Convex Optimization with Unbounded Gradients](https://mlsanity.com/paper/2605.15522/) (2026)
- [Adam Converges in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2606.22326/) (2026)
- [Optimizing the Preconditioner: A Black-box Online-to-Nonconvex Conversion with Static Regret Minimization Oracles](https://mlsanity.com/paper/2607.17607/) (2026)
- [The Exponential Price of Determinism in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2609.23837/) (2026)
- [Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints](https://mlsanity.com/paper/2610.13787/) (2026)

## Abstract (arXiv)

In this work, we offer a theoretical analysis of two modern optimization techniques for training large and complex models: (i) adaptive optimization algorithms, such as Adam, and (ii) the model exponential moving average (EMA). Specifically, we demonstrate that a clipped version of Adam with model EMA achieves the optimal convergence rates in various nonconvex optimization settings, both smooth and nonsmooth. Moreover, when the scale varies significantly across different coordinates, we demonstrate that the coordinate-wise adaptivity of Adam is provably advantageous. Notably, unlike previous analyses of Adam, our analysis crucially relies on its core elements -- momentum and discounting factors -- as well as model EMA, motivating their wide applications in practice.