Muon in Associative Memory Learning: Training Dynamics and Scaling Laws

2026-02-05

AI Summary

This work addresses the slow convergence of conventional gradient descent on low-frequency components in associative memory tasks with long-tailed frequency distributions, which severely limits training efficiency. Within a linear associative memory framework, the authors systematically analyze the training dynamics and scaling behavior of the Muon optimizer, theoretically demonstrating for the first time that it constructs an implicit adaptive preconditioner through sign-based matrix gradient updates, thereby achieving task-aligned optimization. This mechanism yields exponential acceleration in noiseless settings and superior scaling efficiency under noisy conditions. Through spectral frequency analysis, preconditioning theory, and experiments on both synthetic benchmarks and LLaMA-style pretraining, the study validates Muon’s pronounced advantages in long-tailed classification and language model pretraining.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax retrieval and a hierarchical frequency spectrum over query-answer pairs, with and without label noise. In this setting, we show that Gradient Descent (GD) learns frequency components at highly imbalanced rates, leading to slow convergence bottlenecked by low-frequency components. In contrast, the Muon optimizer mitigates this imbalance, leading to faster and more uniform progress. Specifically, in the noiseless case, Muon achieves an exponential speedup over GD; in the noisy case with a power-decay frequency spectrum, we derive Muon's optimization scaling law and demonstrate its superior scaling efficiency over GD. Furthermore, we show that Muon can be interpreted as an implicit matrix preconditioner arising from adaptive task alignment and block-symmetric gradient structure. In contrast, the preconditioner with coordinate-wise sign operator could match Muon under oracle access to unknown task representations, which is infeasible for SignGD in practice. Experiments on synthetic long-tail classification and LLaMA-style pre-training corroborate the theory.

Problem

• associative memory
• training dynamics
• scaling laws
• frequency spectrum
• optimization

Innovation

• Muon optimizer
• matrix sign gradient
• frequency imbalance
• implicit preconditioning
• scaling laws