CacheMuon: Using Temporal Preconditioning To Approximate Polar Factor

2026-06-15 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

CacheMuon is a novel optimization method designed to enhance the efficiency of the Muon optimizer by addressing the computational cost of its polar factor calculation. Muon, which computes updates using the polar factor of the momentum matrix, relies on an expensive Newton-Schulz iteration at every optimization step. Recognizing that the momentum matrix changes smoothly over training, CacheMuon introduces a temporal preconditioning technique. This method reuses information from prior optimization steps to approximate the polar factor at the current step, thereby reducing redundant orthogonalization FLOPs across iterations. Analyzed as an inexact Muon update, CacheMuon's error is managed by fresh-solver error and cache staleness. Empirically, it offers a controllable quality-efficiency trade-off: conservative settings achieve performance comparable to fresh Muon in language-model and vision training with reduced orthogonalization FLOPs, while aggressive settings provide greater arithmetic savings at the cost of minor validation-quality degradation.

Key takeaway

For Machine Learning Engineers optimizing large model training, CacheMuon offers a critical trade-off for computational efficiency. If you are struggling with the high cost of Muon's polar factor computations, you should evaluate CacheMuon's temporal preconditioning. You can achieve near-identical performance to Muon with reduced FLOPs using conservative settings, or opt for greater arithmetic savings with aggressive settings, accepting modest validation-quality degradation. This allows you to fine-tune your training process for specific resource constraints or performance targets.

Key insights

CacheMuon improves optimizer efficiency by temporally preconditioning polar factor computations, reusing past data to reduce redundant orthogonalization.

Principles

• Temporal correlation enables reuse in optimization.
• Inexact updates can balance quality and efficiency.
• Cache staleness impacts approximation error.

Method

CacheMuon approximates the polar factor by reusing information from previous optimization steps through temporal preconditioning, reducing redundant orthogonalization computation across iterations.

In practice

• Apply conservative thresholds for high fidelity.
• Use aggressive thresholds for maximum FLOPs savings.
• Evaluate quality-efficiency frontier for specific tasks.

Topics

• Optimization Algorithms
• Machine Learning Optimizers
• Temporal Preconditioning
• Polar Factor Approximation
• Computational Efficiency
• Language Models
• Vision Training

Best for: AI Engineer, NLP Engineer, Computer Vision Engineer, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.