# SoftSignSGD(S3): An Enhanced Optimizer for Practical DNN Training and Loss Spikes Minimization Beyond Adam

arXiv: https://arxiv.org/abs/2507.06464 · HTML: https://mlsanity.com/paper/2507.06464/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 58 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [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)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [GLM-130B: An Open Bilingual Pre-trained Model](https://mlsanity.com/paper/2210.02414/) (2022)
- [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)
- [ConvNeXt V2: Co-designing and Scaling ConvNets with Masked Autoencoders](https://mlsanity.com/paper/2301.00808/) (2023)
- [Baichuan 2: Open Large-scale Language Models](https://mlsanity.com/paper/2309.10305/) (2023)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [Understanding the Difficulty of Training Transformers](https://mlsanity.com/paper/2004.08249/) (2020)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)

## Built on by (most cited first, 1 of 1)

- [Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations](https://mlsanity.com/paper/2604.12968/) (2026)

## Abstract (arXiv)

Adam has proven remarkable successful in training deep neural networks, but the mechanisms underlying its empirical successes and limitations remain underexplored. In this study, we demonstrate that the effectiveness of Adam stems largely from its similarity to SignSGD in robustly handling large gradient fluctuations, yet it is also vulnerable to destabilizing loss spikes due to its uncontrolled update scaling. To enhance the advantage of Adam and mitigate its limitation, we propose SignSoftSGD (S3), a novel optimizer with three key innovations. First, S3 generalizes the sign-like update by employing a flexible p-th order momentum (p ≥ 1) in the denominator, departing from the conventional second-order momentum (variance) preconditioning. This design enables enhanced performance while achieving stable training even with aggressive learning rates. Second, S3 minimizes the occurrences of loss spikes through unified exponential moving average coefficients for numerator and denominator momenta, which inherently bound updates to [-1, 1] and simplify hyperparameter tuning. Third, S3 incorporates an equivalent Nesterov's accelerated gradient(NAG) module, accelerating convergence without memory overhead. Theoretically, we prove that S3 achieves the optimal convergence rate of Oleft(frac1T^sfrac14right) for general nonconvex stochastic optimization under weak assumptions. Extensive experiments across a range of vision and language tasks show that textsfsmall S3 not only converges more rapidly and improves performance but also rarely experiences loss spikes, even with a textbfbm10 × larger learning rate. In fact, S3 delivers performance comparable to or better than AdamW with 2 × the training steps, establishing its efficacy in both efficiency and final task performance.