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TL;DR: We introduce SoftSignum and SoftMuon, smoothly relaxing sign-based optimization. Supported by a unified geometry-relaxation theory and non-convex convergence proofs, they resolve parameter heterogeneity and outperform baselines.

This repository contains the official implementation for the paper "Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling"

The repository provides:

• The official implementation of SoftSignum method.
• The official implementation of SoftMuon for different distributed training frameworks.

Abstract:

Sign-based and LMO-inspired optimizers have recently attracted substantial attention in deep learning due to their strong performance and low memory footprint. However, their fixed-magnitude updates can hurt terminal convergence: they decouple update mechanisms from gradient magnitudes and fail to account for parameter heterogeneity, often leading to oscillation rather than convergence. We propose SoftSignum, a smooth relaxation of sign-based optimization that replaces the hard sign map with a temperature-controlled soft-sign transformation, enabling a parameter-wise transition from sign-like updates to magnitude-sensitive SGD-like steps. We complement it with an adaptive quantile-based temperature schedule and extend the same principle to matrix-valued optimizers, obtaining SoftMuon. We also develop a generalized geometry-relaxation framework based on strongly convex regularizers and Fenchel conjugates, proving convergence in stochastic non-convex setting. Experiments on diverse deep learning tasks, including LLM pretraining, show that SoftSignum and SoftMuon consistently improve over their hard sign-based counterparts and standard AdamW.

Using SoftSignum or SoftMuon in practice

To use SoftSignum outside of this repository, you need only the following:

• softsignum.py: the minimal single-file implementation;
• the hyperparameters section.

To use SoftMuon outside of this repository, you need only the following:

• Choose a suitable implementation:
  • softmuon.py: based on Keller Jordan's reference Muon implementation;
  • d-softmuon.py: based on toothacher17's Muon implementation for Megatron-LM;
  • dion_softmuon.py: based on Microsoft's Muon implementation used in the Dion project;
• The section about hyperparameters.

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Table of contents

• Using SoftSignum or SoftMuon in practice
• Hyperparameters
• How to cite

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Hyperparameters

SoftSignum and SoftMuon introduce only a small number of additional hyperparameters:

• the transition point $\alpha_{\text{sign}}$;
• the saturation tolerance $\varepsilon$;
• the number of Newton iterations $N_q$ used for quantile computation.

In our experiments, we use these standard values:

$\alpha_{\text{sign}} = 0.9,\quad \varepsilon = 10^{-4},\quad N_q = 10.$

These values provide a simple default configuration and allow SoftSignum and SoftMuon to be integrated into existing Signum and Muon pipelines without additional tuning.

Moreover, we investigate the robustness of our method with respect to the important hyperparameter $\alpha_{\text{sign}}$ and show that, when varying this hyperparameter over the range from 0.9 to 0.3, the final metrics remain stable.

How to cite

@misc{feoktistov2026softsign,
      title={Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling}, 
      author={Dmitrii Feoktistov and Timofey Belinsky and Andrey Veprikov and Amir Zainullin and Aleksandr Beznosikov},
      year={2026},
      eprint={2605.31371},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}