This work adapt and enhance preconditioned gradient methods to overcome the practical challenges of large-scale LLM pretraining and proposes algorithmic modifications including per-step QR orthogonalization and improved preconditioning strategies to enable stable training in these regimes.
Abstract
Higher-order optimizers such as Muon and SOAP offer faster convergence than AdamW, but their computational cost and numerical stability challenges have limited adoption at scale. In this work, we adapt and enhance preconditioned gradient methods to overcome the practical challenges of large-scale LLM pretraining. We first identify instabilities in SOAP at large batch sizes and propose algorithmic modifications including per-step QR orthogonalization and improved preconditioning strategies that eliminate loss spikes and enable stable training in these regimes. We then present a unified empirical study of SOAP, Muon, and AdamW using update-RMS matching to ensure fair learning rate transfer across optimizers. As part of this analysis, we empirically evaluate the orthogonalization quality of Muon. Our experiments on multi-billion-parameter models trained on trillions of tokens reveal that SOAP and Muon consistently outperform AdamW at the scales we tested. Notably, at batch sizes of up to 100M tokens for next-token prediction, these optimizers maintain training stability and quality while AdamW degrades. To enable efficient training at large scale, we introduce a layer-wise distributed optimizer compatible with Megatron-LM. Our implementation balances memory and hides communication while avoiding approximations to the optimizer computations, thus retaining their convergence benefits. Additionally, we identify and build specific system-level improvements to further accelerate our layer-wise implementation. To support the research community, we release a codebase that contains emerging algorithms for optimization: https://github.com/NVIDIA-NeMo/Emerging-Optimizers
Spectral-Aware Muon is introduced, which holds the head at the Muon scale and amplifies the bulk using a static spectral prior, and both variants outperform tuned AdamW and Muon (Scion implementation) baselines in all evaluated model-scale and batch-size configurations.
Xiaodong Wu, Wenyi Yu, Chao Zhang et al.· 1 citation
Periodic Row-wise Muon is introduced, which performs a full NS5 spectral update once every \(K\) steps and applies a low compute and communication cost row-wise constrained update based on the current momentum at the remaining steps to preserve Muon's generative quality advantage while translating it into end-to-end training efficiency for large DiTs.
Chenghao Li, Xiao Han, Xin-Xin Huang et al.· 0 citations
We investigate how optimizers scale across the overtraining axis and show that relative optimizer performance and optimal hyperparameters change substantially with training horizon. In particular, we study how matrix-preconditioned methods (Muon and SOAP) and a momentum-scheduled method (ADANA) scale relative to AdamW. We compare these four optimizers across models from 51M to 253M parameters and overtraining (OT) factors from 1x to 256x, sweeping the base learning rate at every setting. The preferred learning rate schedule can reverse across the overtraining axis, the best weight decay coefficient scales approximately as sqrt(OT), and longer horizons generally favor longer fixed memory. ADANA's scaling advantage over AdamW persists after tuning AdamW's fixed memory separately at each horizon. Log-time weight decay and momentum cooldown provide substantial gains for ADANA that compound as training increases. With this treatment, ADANA outscales AdamW with an exponent advantage close to that predicted by DANA theory on power-law random features. Muon and SOAP instead provide roughly constant token-efficiency advantages over AdamW across most of the measured range, although SOAP may gain further at the highest overtraining factors. ADANA begins behind both matrix-preconditioned optimizers but closes the gaps as training increases, surpassing Muon and becoming competitive with SOAP at our highest OT factors. These results establish training horizon as an essential axis for optimizer evaluation and design.
PoLoRA is introduced, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates.
Nikhil Ghosh, Tetiana Parshakova, Robert M. Gower· arXiv.org· 1 citation
Muon has emerged as a highly effective optimizer for large language model training, often achieving superior convergence and performance compared with the widely adopted Adam and AdamW optimizers. Nevertheless, Muon is prone to training instability due to its spectral flattening, manifested by loss spikes and unbounded growth of model weights. Existing approaches primarily rely on weight or attention-logit clipping, which require architecture-specific modifications and do not directly address instability across all model components. We propose MomentUm SpEctral Clipping (Musec), which replaces Muon's spectral flattening with spectral clipping: rather than setting all singular values of the momentum matrix to approximately one, Musec clips singular values that exceed a threshold while preserving the underlying spectral structure of the momentum. Our strategy provides an optimizer-level, architecture-agnostic mechanism for stabilizing Muon training. We further develop Soft Musec, an efficient implementation that uses a smooth spectral saturation function approximated by coupled Newton-Schulz iterations. Theoretically, we establish convergence guarantees for Musec in nonconvex nonsmooth stochastic optimization. To the best of our knowledge, this is the first convergence guarantee for Muon-type methods in the nonconvex nonsmooth setting. We provide empirical studies to show that Soft Musec consistently improves training stability over existing Muon variants across a wide range of learning rates and model sizes. Notably, Soft Musec remains stable in settings where existing Muon variants diverge, while matching their performance under well-tuned configurations.
Zhuanghua Liu, Menglian Wang, Luo Luo· 0 citations
Method is introduced, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling, and is positioned as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.
Guang-Yuan Wang, Mads Toftrup, Sebastian Loeschcke et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
