# A Sufficient Condition for Convergences of Adam and RMSProp

arXiv: https://arxiv.org/abs/1811.09358 · HTML: https://mlsanity.com/paper/1811.09358/

## Key facts

- Cited by 126 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2018.

## Built on by (most cited first, 30 of 126)

- [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)
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- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [Model Agnostic Sample Reweighting for Out-of-Distribution Learning](https://mlsanity.com/paper/2301.09819/) (2023)
- [High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize](https://mlsanity.com/paper/2204.02833/) (2022)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Convergence of Muon with Newton-Schulz](https://mlsanity.com/paper/2601.19156/) (2026)
- [Learning with Multiclass AUC: Theory and Algorithms](https://mlsanity.com/paper/2107.13171/) (2021)
- [AdaTask: A Task-aware Adaptive Learning Rate Approach to Multi-task Learning](https://mlsanity.com/paper/2211.15055/) (2022)
- [Provable Adaptivity of Adam under Non-uniform Smoothness](https://mlsanity.com/paper/2208.09900/) (2022)
- [Adaptivity without Compromise: A Momentumized, Adaptive, Dual Averaged Gradient Method for Stochastic Optimization](https://mlsanity.com/paper/2101.11075/) (2021)
- [Probabilistic Bilevel Coreset Selection](https://mlsanity.com/paper/2301.09880/) (2023)
- [Understanding Adam Optimizer via Online Learning of Updates: Adam is FTRL in Disguise](https://mlsanity.com/paper/2402.01567/) (2024)
- [Towards Practical Adam: Non-Convexity, Convergence Theory, and Mini-Batch Acceleration](https://mlsanity.com/paper/2101.05471/) (2021)
- [Toward Communication Efficient Adaptive Gradient Method](https://mlsanity.com/paper/2109.05109/) (2021)
- [Two Sides of One Coin: the Limits of Untuned SGD and the Power of Adaptive Methods](https://mlsanity.com/paper/2305.12475/) (2023)
- [Efficient Federated Learning via Local Adaptive Amended Optimizer with Linear Speedup](https://mlsanity.com/paper/2308.00522/) (2023)
- [Adam with model exponential moving average is effective for nonconvex optimization](https://mlsanity.com/paper/2405.18199/) (2024)
- [Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity](https://mlsanity.com/paper/2410.08198/) (2024)
- [Adam-family Methods for Nonsmooth Optimization with Convergence Guarantees](https://mlsanity.com/paper/2305.03938/) (2023)
- [AdaSAM: Boosting Sharpness-Aware Minimization with Adaptive Learning Rate and Momentum for Training Deep Neural Networks](https://mlsanity.com/paper/2303.00565/) (2023)
- [On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions](https://mlsanity.com/paper/2402.03982/) (2024)
- [Facial Emotion Recognition: State of the Art Performance on FER2013](https://mlsanity.com/paper/2105.03588/) (2021)

## Abstract (arXiv)

Adam and RMSProp are two of the most influential adaptive stochastic algorithms for training deep neural networks, which have been pointed out to be divergent even in the convex setting via a few simple counterexamples. Many attempts, such as decreasing an adaptive learning rate, adopting a big batch size, incorporating a temporal decorrelation technique, seeking an analogous surrogate, etc., have been tried to promote Adam/RMSProp-type algorithms to converge. In contrast with existing approaches, we introduce an alternative easy-to-check sufficient condition, which merely depends on the parameters of the base learning rate and combinations of historical second-order moments, to guarantee the global convergence of generic Adam/RMSProp for solving large-scale non-convex stochastic optimization. Moreover, we show that the convergences of several variants of Adam, such as AdamNC, AdaEMA, etc., can be directly implied via the proposed sufficient condition in the non-convex setting. In addition, we illustrate that Adam is essentially a specifically weighted AdaGrad with exponential moving average momentum, which provides a novel perspective for understanding Adam and RMSProp. This observation coupled with this sufficient condition gives much deeper interpretations on their divergences. At last, we validate the sufficient condition by applying Adam and RMSProp to tackle a certain counterexample and train deep neural networks. Numerical results are exactly in accord with our theoretical analysis.