# Convergence Guarantees for RMSProp and Adam in Generalized-smooth Non-convex Optimization with Affine Noise Variance

arXiv: https://arxiv.org/abs/2404.01436 · HTML: https://mlsanity.com/paper/2404.01436/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 47 references that are in the map.
- First posted on arXiv 1 Apr 2024.

## Builds on (most cited first, 30 of 35)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [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)
- [Regularizing and Optimizing LSTM Language Models](https://mlsanity.com/paper/1708.02182/) (2017)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [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)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [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)
- [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)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance](https://mlsanity.com/paper/2202.05791/) (2022)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD](https://mlsanity.com/paper/2302.06570/) (2023)
- [Convergence guarantees for RMSProp and ADAM in non-convex optimization and an empirical comparison to Nesterov acceleration](https://mlsanity.com/paper/1807.06766/) (2018)
- [Non-convex Distributionally Robust Optimization: Non-asymptotic Analysis](https://mlsanity.com/paper/2110.12459/) (2021)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)
- [Generalized-Smooth Nonconvex Optimization is As Efficient As Smooth Nonconvex Optimization](https://mlsanity.com/paper/2303.02854/) (2023)
- [The Complexity of Making the Gradient Small in Stochastic Convex Optimization](https://mlsanity.com/paper/1902.04686/) (2019)

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

- [Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under (L0, L1)-Smoothness](https://mlsanity.com/paper/2502.07923/) (2025)
- [Refining Adaptive Zeroth-Order Optimization at Ease](https://mlsanity.com/paper/2502.01014/) (2025)
- [Subspace Networks: Scaling Decentralized Training with Communication-Efficient Model Parallelism](https://mlsanity.com/paper/2506.01260/) (2025)
- [On the O(frac√dK1/4) Convergence Rate of AdamW Measured by ℓ1 Norm](https://mlsanity.com/paper/2505.11840/) (2025)
- [MGDA Converges under Generalized Smoothness, Provably](https://mlsanity.com/paper/2405.19440/) (2024)
- [Adaptive Gradient Normalization and Independent Sampling for (Stochastic) Generalized-Smooth Optimization](https://mlsanity.com/paper/2410.14054/) (2024)
- [On the Interaction of Batch Noise, Adaptivity, and Compression, under (L0,L1)-Smoothness: An SDE Approach](https://mlsanity.com/paper/2506.00181/) (2025)
- [Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method](https://mlsanity.com/paper/2603.18899/) (2026)
- [Can Adaptive Gradient Methods Converge under Heavy-Tailed Noise? A Case Study of AdaGrad](https://mlsanity.com/paper/2605.18694/) (2026)
- [A unified convergence theory for adaptive first-order methods in the nonconvex case, including AdaNorm, full and diagonal AdaGrad and Muon](https://mlsanity.com/paper/2604.17423/) (2026)
- [MGUP: A Momentum-Gradient Alignment Update Policy for Stochastic Optimization](https://mlsanity.com/paper/2606.17526/) (2026)
- [Adam Converges in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2606.22326/) (2026)
- [Convergence rates for the RMSprop optimizer with full control of the hyperparameters](https://mlsanity.com/paper/2608.30382/) (2026)
- [Adam under Generalized Smoothness with Second-Moment-Type Stochastic Gradients](https://mlsanity.com/paper/2609.37787/) (2026)

## Abstract (arXiv)

This paper provides the first tight convergence analyses for RMSProp and Adam in non-convex optimization under the most relaxed assumptions of coordinate-wise generalized smoothness and affine noise variance. We first analyze RMSProp, which is a special case of Adam with adaptive learning rates but without first-order momentum. Specifically, to solve the challenges due to dependence among adaptive update, unbounded gradient estimate and Lipschitz constant, we demonstrate that the first-order term in the descent lemma converges and its denominator is upper bounded by a function of gradient norm. Based on this result, we show that RMSProp with proper hyperparameters converges to an ε-stationary point with an iteration complexity of mathcal O(ε-4). We then generalize our analysis to Adam, where the additional challenge is due to a mismatch between the gradient and first-order momentum. We develop a new upper bound on the first-order term in the descent lemma, which is also a function of the gradient norm. We show that Adam with proper hyperparameters converges to an ε-stationary point with an iteration complexity of mathcal O(ε-4). Our complexity results for both RMSProp and Adam match with the complexity lower bound established in citearjevani2023lower.