T-MUON: Tensorizing Muon via t-SVD for Large Language Model Pre-training

The success of large language models (LLMs) has been accompanied by continued growth in model size and pretraining costs. Muon offers high accuracy and training efficiency in LLM pretraining. Recent work introduces row-wise normalization into Muon to balance update magnitudes and improve pretraining performance. However, row-wise normalization alone cannot accommodate different imbalance patterns in update matrices. In this paper, we propose an improved Muon optimizer, called matrix-equilibrating Muon~(MeqMuon), for LLM pretraining. MeqMuon balances both row and column magnitudes through normalization that can be automatically tailored to different imbalance patterns without manual intervention. Moreover, MeqMuon eliminates the need to store AdamW's second-moment estimates, reducing optimizer-state memory usage. Empirical results demonstrate that MeqMuon achieves better convergence performance than AdamW, Muon, and other baselines in LLM pretraining.

Orthogonalized-momentum optimizers such as Muon improve transformer training by approximately whitening/orthogonalizing matrix-valued momentum updates via a short polar-decomposition iteration. However, polar-factor approximations typically require multiple large matrix multiplications, and the resulting overhead can be substantial and hardware-dependent. We introduce MUD (MomentUm Decorrelation), a complementary whitening approach that replaces Muon's polar update with a triangular (Cholesky-like) whitening surrogate inspired by classical Gram--Schmidt and Gauss-Seidel ideas. We show that row-orthonormal matrices are fixed points of the MUD map, relate the inner step to symmetric Gauss-Seidel preconditioning of the Gram matrix, and prove quadratic local convergence near the fixed point. In terms of time-to-perplexity, MUD yields consistent 10-50% wall-clock improvements over tuned AdamW and Muon in time-to-perplexity, typically converging slightly slower per step than Muon but with substantially lower optimizer overhead -- relative to Muon, MUD improves peak tokens/s by roughly $1.3-2.6\times$ across most settings and up to nearly $3\times$ on GPT-2 large on an A100. We also demonstrate training a ESM-2 150M protein language model, where MUD matches Muon-level validation perplexity in significantly less wall-clock time.

Muon improves large-scale training by applying a spectral-norm steepest-descent update to matrix parameters, but practical models also contain parameter blocks that do not fit dense-matrix geometry. One important case is the tied vocabulary table, which appears in language models and other token generators and can receive multiple structurally different gradient sources, from sparse input lookups to dense output-classifier updates. In the reference recipe these blocks are handed to an auxiliary AdamW optimizer, which restores second-moment state and updates the aliased table as a generic tensor. We propose AF-Muon, an AdamW-free extension of Muon that keeps the Muon matrix update for hidden weight matrices while using a support-aware finite-cap linear minimization oracle for tied vocabulary tables and an RMS-normalized update for one-dimensional auxiliary parameters. AF-Muon therefore trains every parameter class with a single first-moment buffer and no second-moment state, saving around 20% optimizer-state memory relative to Hybrid Muon in our benchmark. Across nine tied-token settings - decoder-only language models from 124M to 1B parameters, a fully shared T5-style encoder-decoder, and ImageGPT-style image-token, protein, and sparse-MoE variants, spanning text, image, and protein-sequence data - AF-Muon improves mean validation loss and perplexity over both Hybrid Muon and a SCION-style Sign endpoint. Long-horizon runs and hyperparameter sensitivity studies confirm the gain is robust, and identical-momentum diagnostics attribute it to the finite cap, which preserves more within-row magnitude than Sign while bounding the coordinate concentration of row-RMS. These results identify tied vocabulary tables as a distinct optimizer geometry and yield a robust AdamW-free Muon variant across models, modalities, and architectures, with about 1% step-time overhead in matched training.