# Adam Converges in Nonsmooth Nonconvex Optimization

arXiv: https://arxiv.org/abs/2606.22326 · HTML: https://mlsanity.com/paper/2606.22326/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 54 references that are in the map.
- First posted on arXiv 21 Jun 2026.

## Builds on (most cited first, 30 of 36)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (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)
- [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)
- [A Novel Convergence Analysis for Algorithms of the Adam Family](https://mlsanity.com/paper/2112.03459/) (2021)
- [Convergence and Dynamical Behavior of the ADAM Algorithm for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1810.02263/) (2018)
- [Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion](https://mlsanity.com/paper/2302.03775/) (2023)
- [In Search of Adam's Secret Sauce](https://mlsanity.com/paper/2505.21829/) (2025)
- [Convergence guarantees for RMSProp and ADAM in non-convex optimization and an empirical comparison to Nesterov acceleration](https://mlsanity.com/paper/1807.06766/) (2018)
- [Oracle Complexity in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2104.06763/) (2021)
- [A gradient sampling method with complexity guarantees for Lipschitz functions in high and low dimensions](https://mlsanity.com/paper/2112.06969/) (2021)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [On the Finite-Time Complexity and Practical Computation of Approximate Stationarity Concepts of Lipschitz Functions](https://mlsanity.com/paper/2112.09002/) (2021)
- [Provable Adaptivity of Adam under Non-uniform Smoothness](https://mlsanity.com/paper/2208.09900/) (2022)
- [Deterministic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2302.08300/) (2023)
- [Understanding Adam Optimizer via Online Learning of Updates: Adam is FTRL in Disguise](https://mlsanity.com/paper/2402.01567/) (2024)
- [Adam with model exponential moving average is effective for nonconvex optimization](https://mlsanity.com/paper/2405.18199/) (2024)
- [A Generalized Online Mirror Descent with Applications to Classification and Regression](https://mlsanity.com/paper/1304.2994/) (2013)
- [Random Scaling and Momentum for Non-smooth Non-convex Optimization](https://mlsanity.com/paper/2405.09742/) (2024)
- [Online Convex Optimization with Heavy Tails: Old Algorithms, New Regrets, and Applications](https://mlsanity.com/paper/2508.07473/) (2025)
- [No Dimension-Free Deterministic Algorithm Computes Approximate Stationarities of Lipschitzians](https://mlsanity.com/paper/2210.06907/) (2022)
- [General framework for online-to-nonconvex conversion: Schedule-free SGD is also effective for nonconvex optimization](https://mlsanity.com/paper/2411.07061/) (2024)
- [Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance](https://mlsanity.com/paper/2404.01436/) (2024)
- [On the Complexity of Finding Small Subgradients in Nonsmooth Optimization](https://mlsanity.com/paper/2209.10346/) (2022)

## Abstract (arXiv)

Adam is one of the most widely implemented and influential modern optimizers. Why is it effective across different optimization problems in practice? This question arguably lies at the center of the optimization community over the last decade and has motivated a substantial body of work aimed at understanding its convergence behavior. However, existing studies have mainly focused on the convergence rate of Adam in smooth nonconvex optimization, which unfortunately does not adequately capture practical settings, since many real-world problems are nonsmooth, such as those arising in training neural networks. Thus, these studies cannot fully explain the popularity and empirical success of Adam. Recently, an insightful and powerful framework called Online-to-Nonconvex Conversion has opened a new way to analyze Adam for nonsmooth nonconvex optimization. Unfortunately, prior works along this line share two common limitations. First, all of them ignore the important bias-correction term in the original Adam algorithm. Second and more importantly, many of them require extra operations that are not used in Adam, such as a clipping step. Therefore, the convergence guarantee for the original Adam method still remains unclear. In this work, we present the first finite-time analysis for the classical form of Adam, i.e., with the bias-correction step and without further algorithmic modifications, and prove that a randomly scaled learning rate ensures a convergence rate of 1/T^frac213 for nonsmooth nonconvex optimization. Moreover, our result provably applies to the modern heavy-tailed noise regime, which is closer to practice. Interestingly, our theory is established under the parameter choice β1=β2, aligning with the recent empirical studies.