Organizations: King Abdullah University of Science and Technology · Arizona State University
Abstract
Muon and related normalized optimizers decouple the choice of update direction from the choice of step scale, but their practical performance remains sensitive to the scale of the normalized step. We study adaptive scaling rules for Muon in general norm geometries and develop three complementary algorithms. For smooth non-convex objectives, we introduce Distance-Adaptive Muon, whose trust-region radius is set from the radius explored by the trajectory, and prove a stationarity guarantee under a bounded-trajectory assumption. We then turn to star-convex objectives, a tractable model of the favorable global geometry often used to reason about the empirical loss landscapes of deep neural networks, where objective-gap guarantees are possible. In this setting, we first introduce Scale-Calibrated Muon, which keeps Muon's exponential moving average but sets the step length from a local descent certificate computed from the current gradient and momentum. For this method, we prove a last-iterate O(1/T) objective-gap bound under a bounded initial sublevel-set assumption, where the corresponding radius parameter appears only in the analysis and not in the algorithm. Finally, we develop Distance-Free Muon, a recentered trust-region method that uses a scalar distance certificate and a majorized one-dimensional search to select the trust-region radius without requiring the unknown distance from the initialization to a global minimizer. Experiments on Transformer language modeling (GPT-124M/WikiText-103) and image classification (ViT-Tiny/CIFAR-100) show that the proposed adaptive scaling rules reduce sensitivity to manual scale tuning and match or improve tuned fixed-scale Muon baselines under the tested budgets.
Muon fixes the \emph{direction} of every matrix-valued update at the polar factor of its momentum, while each layer's step \emph{magnitude} is addressed only by a static shape correction. We derive a dynamic per-layer scalar by adapting the LARS/LAMB trust-ratio principle to the orthogonalized setting, where the standard denominator candidates---the raw momentum norm or the polar-factor norm---either live in the wrong unit space or carry no update-scale information. The resulting method, \emph{OrScale}, uses the norm of the parameter-space direction actually applied and anchors each layer's ratio at one via a per-layer calibration, so that the Moonlight recipe (tuned for AdamW, shared with Muon via RMS matching) transfers with \emph{no additional sweep}; a component ablation confirms each design choice is individually load-bearing. Theoretically, OrScale retains a nuclear-norm O(1/T) convergence rate for any clipped multiplier and achieves a strict layer-adaptive descent gain κeff>1 under two conditions estimable from standard training diagnostics---a bound that predicts the gain should \emph{grow with architectural heterogeneity}. Experiments confirm the prediction: with every hyperparameter inherited verbatim from the Moonlight recipe, OrScale matches or beats Muon+Moonlight across dense 125M--1.1B FineWeb-Edu pre-training, and on a 16B-A3B mixture-of-experts model---where the logged trust ratios separate cleanly by layer class---the gap widens by an order of magnitude to 0.130 nats (3.8% relative) at parity wall-clock cost.
The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become trapped in sharp local minima. In this work, we present MONA, an optimizer that bridges Muon's orthogonalization framework with curvature-aware acceleration. MONA adds an acceleration term directly into Muon's gradient processing pipeline. This term is calculated from the exponential moving average of gradient differences. We provide a detailed convergence analysis for MONA, showing that the acceleration term enables escape from sharp minima while preserving Muon's spectral-norm regularization. Empirically, MONA achieves better convergence and downstream task performance compared to both Muon and AdamW across three scales of Mixture-of-Experts pretraining, spanning from 1B to 68B parameters, with the largest model trained on 1 trillion tokens. Furthermore, we conduct supervised fine-tuning on the MOE-68B-A3B model and evaluate it on general capability, mathematical reasoning, and code generation benchmarks, where MONA achieves SOTA performance.
Optimizer design plays a central role in efficient language model pretraining, directly affecting optimization dynamics, convergence speed, and compute cost under fixed training budgets. Muon has emerged as a strong optimizer by orthogonalizing momentum updates, yielding a matrix-valued analogue of sign-based normalization. However, unlike Adam-style methods, Muon does not explicitly incorporate gradient-variance information into its updates. Motivated by Adam's variance-adaptive interpretation, we propose Muon-NSR and Muon-VS, two variance-adaptive Muon variants for language model pretraining. Muon-NSR applies noise-to-signal ratio (NSR) modulation before Newton--Schulz orthogonalization, whereas Muon-VS uses variance scaling (VS) without introducing any additional hyperparameters beyond those of Muon. Both methods preserve Muon's spectral normalization structure while requiring only one additional variance buffer. Experiments on Llama-style and GPT-2 pretraining across model scales from 125M to 1.2B parameters show that our methods improve over well-tuned Muon baselines and remain competitive with representative adaptive Muon-family baselines. On Llama-1.2B, Muon-VS achieves a 1.33× step-to-target speedup over a well-tuned Muon baseline, with Muon's final validation loss as the target. These results indicate that variance-adaptive modulation is a simple and effective mechanism for improving Muon-style optimizers in language model pretraining.