Muon$^2$: Boosting Muon via Adaptive Second-Moment Preconditioning

• URL: http://arxiv.org/abs/2604.09967v1
• Date: Sat, 11 Apr 2026 00:27:40 GMT
• Title: Muon$^2$: Boosting Muon via Adaptive Second-Moment Preconditioning
• Abstract summary: Muon has emerged as a promising computation for large-scale foundation model pre-training.<n>We propose Muon$2$, a memory-efficient factorized variant that preserves most of the gains of Muon$2$ with negligible memory overhead.
• Score: 18.570226339282296
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Muon has emerged as a promising optimizer for large-scale foundation model pre-training by exploiting the matrix structure of neural network updates through iterative orthogonalization. However, its practical efficiency is limited by the need for multiple Newton--Schulz (NS) iterations per optimization step, which introduces non-trivial computation and communication overhead. We propose Muon$^2$, an extension of Muon that applies Adam-style adaptive second-moment preconditioning before orthogonalization. Our key insight is that the core challenge of polar approximation in Muon lies in the ill-conditioned momentum matrix, of which the spectrum is substantially improved by Muon$^2$, leading to faster convergence toward a practically sufficient orthogonalization. We further characterize the practical orthogonalization quality via directional alignment, under which Muon$^2$ demonstrates dramatic improvement over Muon at each polar step. Across GPT and LLaMA pre-training experiments from 60M to 1.3B parameters, Muon$^2$ consistently outperforms Muon and recent Muon variants while reducing NS iterations by 40\%. We further introduce Muon$^2$-F, a memory-efficient factorized variant that preserves most of the gains of Muon$^2$ with negligible memory overhead.

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