# Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case

arXiv: https://arxiv.org/abs/2307.11782 · HTML: https://mlsanity.com/paper/2307.11782/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 82 references that are in the map.
- First posted on arXiv 20 Jul 2023.

## Builds on (most cited first, 30 of 50)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (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)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [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)
- [Stability and Generalization of Learning Algorithms that Converge to Global Optima](https://mlsanity.com/paper/1710.08402/) (2017)
- [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)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [A Unified Convergence Analysis for Shuffling-Type Gradient Methods](https://mlsanity.com/paper/2002.08246/) (2020)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [Variants of RMSProp and Adagrad with Logarithmic Regret Bounds](https://mlsanity.com/paper/1706.05507/) (2017)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [SUPER-ADAM: Faster and Universal Framework of Adaptive Gradients](https://mlsanity.com/paper/2106.08208/) (2021)
- [High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize](https://mlsanity.com/paper/2204.02833/) (2022)
- [On Almost Sure Convergence Rates of Stochastic Gradient Methods](https://mlsanity.com/paper/2202.04295/) (2022)

## Built on by (most cited first, 14 of 14)

- [LDAdam: Adaptive Optimization from Low-Dimensional Gradient Statistics](https://mlsanity.com/paper/2410.16103/) (2024)
- [On the Convergence of Muon and Beyond](https://mlsanity.com/paper/2509.15816/) (2025)
- [MicroAdam: Accurate Adaptive Optimization with Low Space Overhead and Provable Convergence](https://mlsanity.com/paper/2405.15593/) (2024)
- [World Models That Know When They Don't Know - Controllable Video Generation with Calibrated Uncertainty](https://mlsanity.com/paper/2512.05927/) (2025)
- [Convergence rates for the Adam optimizer](https://mlsanity.com/paper/2407.21078/) (2024)
- [On the Convergence of Adam under Non-uniform Smoothness: Separability from SGDM and Beyond](https://mlsanity.com/paper/2403.15146/) (2024)
- [A Comprehensive Framework for Analyzing the Convergence of Adam: Bridging the Gap with SGD](https://mlsanity.com/paper/2410.04458/) (2024)
- [Sharp higher order convergence rates for the Adam optimizer](https://mlsanity.com/paper/2504.19426/) (2025)
- [AdamS: Momentum Itself Can Be A Normalizer for LLM Pretraining and Post-training](https://mlsanity.com/paper/2505.16363/) (2025)
- [Adaptive Regularization for Large-Scale Sparse Feature Embedding Models](https://mlsanity.com/paper/2511.06374/) (2025)
- [MGUP: A Momentum-Gradient Alignment Update Policy for Stochastic Optimization](https://mlsanity.com/paper/2606.17526/) (2026)
- [Unified convergence analysis for gradient descent optimization methods in the training of deep neural networks](https://mlsanity.com/paper/2607.04233/) (2026)
- [OptEMA: Adaptive Exponential Moving Average for Stochastic Optimization with Zero-Noise Optimality](https://mlsanity.com/paper/2603.09923/) (2026)
- [Neural Network Implementation of the Renormalization Group for Fault Diagnosis with Class Imbalance](https://mlsanity.com/paper/2606.18326/) (2026)

## Abstract (arXiv)

Adam is a commonly used stochastic optimization algorithm in machine learning. However, its convergence is still not fully understood, especially in the non-convex setting. This paper focuses on exploring hyperparameter settings for the convergence of vanilla Adam and tackling the challenges of non-ergodic convergence related to practical application. The primary contributions are summarized as follows: firstly, we introduce precise definitions of ergodic and non-ergodic convergence, which cover nearly all forms of convergence for stochastic optimization algorithms. Meanwhile, we emphasize the superiority of non-ergodic convergence over ergodic convergence. Secondly, we establish a weaker sufficient condition for the ergodic convergence guarantee of Adam, allowing a more relaxed choice of hyperparameters. On this basis, we achieve the almost sure ergodic convergence rate of Adam, which is arbitrarily close to o(1/√K). More importantly, we prove, for the first time, that the last iterate of Adam converges to a stationary point for non-convex objectives. Finally, we obtain the non-ergodic convergence rate of O(1/K) for function values under the Polyak-Lojasiewicz (PL) condition. These findings build a solid theoretical foundation for Adam to solve non-convex stochastic optimization problems.