# A Novel Convergence Analysis for Algorithms of the Adam Family

arXiv: https://arxiv.org/abs/2112.03459 · HTML: https://mlsanity.com/paper/2112.03459/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 7 Dec 2021.

## Builds on (most cited first, 23 of 23)

- [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)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [Gradient Sparsification for Communication-Efficient Distributed Optimization](https://mlsanity.com/paper/1710.09854/) (2017)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [On the Linear Speedup Analysis of Communication Efficient Momentum SGD for Distributed Non-Convex Optimization](https://mlsanity.com/paper/1905.03817/) (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)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [Adding Gradient Noise Improves Learning for Very Deep Networks](https://mlsanity.com/paper/1511.06807/) (2015)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [A Sufficient Condition for Convergences of Adam and RMSProp](https://mlsanity.com/paper/1811.09358/) (2018)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions](https://mlsanity.com/paper/1411.3803/) (2014)
- [Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks](https://mlsanity.com/paper/1806.06763/) (2018)
- [A Single Time-Scale Stochastic Approximation Method for Nested Stochastic Optimization](https://mlsanity.com/paper/1812.01094/) (2018)
- [Accelerating Stochastic Composition Optimization](https://mlsanity.com/paper/1607.07329/) (2016)
- [Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization](https://mlsanity.com/paper/1604.03257/) (2016)
- [ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1902.05679/) (2019)
- [Adam^+: A Stochastic Method with Adaptive Variance Reduction](https://mlsanity.com/paper/2011.11985/) (2020)

## Built on by (most cited first, 30 of 50)

- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [Momentum Benefits Non-IID Federated Learning Simply and Provably](https://mlsanity.com/paper/2306.16504/) (2023)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [The Implicit Bias of Adam on Separable Data](https://mlsanity.com/paper/2406.10650/) (2024)
- [GraphFM: Improving Large-Scale GNN Training via Feature Momentum](https://mlsanity.com/paper/2206.07161/) (2022)
- [On the Convergence of Muon and Beyond](https://mlsanity.com/paper/2509.15816/) (2025)
- [Provable Adaptivity of Adam under Non-uniform Smoothness](https://mlsanity.com/paper/2208.09900/) (2022)
- [Understanding Adam Optimizer via Online Learning of Updates: Adam is FTRL in Disguise](https://mlsanity.com/paper/2402.01567/) (2024)
- [Implicit Diffusion: Efficient Optimization through Stochastic Sampling](https://mlsanity.com/paper/2402.05468/) (2024)
- [Two Sides of One Coin: the Limits of Untuned SGD and the Power of Adaptive Methods](https://mlsanity.com/paper/2305.12475/) (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)
- [AdaGrad under Anisotropic Smoothness](https://mlsanity.com/paper/2406.15244/) (2024)
- [On Convergence of Adam for Stochastic Optimization under Relaxed Assumptions](https://mlsanity.com/paper/2402.03982/) (2024)
- [Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case](https://mlsanity.com/paper/2307.11782/) (2023)
- [On the Convergence of Adam under Non-uniform Smoothness: Separability from SGDM and Beyond](https://mlsanity.com/paper/2403.15146/) (2024)
- [Understanding Adam Requires Better Rotation Dependent Assumptions](https://mlsanity.com/paper/2410.19964/) (2024)
- [Divergence Results and Convergence of a Variance Reduced Version of ADAM](https://mlsanity.com/paper/2210.05607/) (2022)
- [TiAda: A Time-scale Adaptive Algorithm for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2210.17478/) (2022)
- [High Probability Convergence of Adam Under Unbounded Gradients and Affine Variance Noise](https://mlsanity.com/paper/2311.02000/) (2023)
- [On the O(frac√dK1/4) Convergence Rate of AdamW Measured by ℓ1 Norm](https://mlsanity.com/paper/2505.11840/) (2025)
- [Fast Adaptive Federated Bilevel Optimization](https://mlsanity.com/paper/2211.01122/) (2022)
- [Enhanced Adaptive Gradient Algorithms for Nonconvex-PL Minimax Optimization](https://mlsanity.com/paper/2303.03984/) (2023)
- [LoCo: Low-Bit Communication Adaptor for Large-scale Model Training](https://mlsanity.com/paper/2407.04480/) (2024)
- [Why Muon Outperforms Adam: A Curvature Perspective](https://mlsanity.com/paper/2606.04662/) (2026)
- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)

## Abstract (arXiv)

Since its invention in 2014, the Adam optimizer has received tremendous attention. On one hand, it has been widely used in deep learning and many variants have been proposed, while on the other hand their theoretical convergence property remains to be a mystery. It is far from satisfactory in the sense that some studies require strong assumptions about the updates, which are not necessarily applicable in practice, while other studies still follow the original problematic convergence analysis of Adam, which was shown to be not sufficient to ensure convergence. Although rigorous convergence analysis exists for Adam, they impose specific requirements on the update of the adaptive step size, which are not generic enough to cover many other variants of Adam. To address theses issues, in this extended abstract, we present a simple and generic proof of convergence for a family of Adam-style methods (including Adam, AMSGrad, Adabound, etc.). Our analysis only requires an increasing or large "momentum" parameter for the first-order moment, which is indeed the case used in practice, and a boundness condition on the adaptive factor of the step size, which applies to all variants of Adam under mild conditions of stochastic gradients. We also establish a variance diminishing result for the used stochastic gradient estimators. Indeed, our analysis of Adam is so simple and generic that it can be leveraged to establish the convergence for solving a broader family of non-convex optimization problems, including min-max, compositional, and bilevel optimization problems. For the full (earlier) version of this extended abstract, please refer to arXiv:2104.14840.