Title: The Polar Express: Optimal Matrix Sign Methods and their Application to the Muon Algorithm
Primary area: optimization
Keywords: polar decomposition, matrix sign, numerical linear algebra, muon, optimization, approximation theory

Abstract: Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine  within the Muon algorithm for training deep neural networks.
However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce *Polar Express*, a new method for computing the polar decomposition. Like Newton–Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it 
very efficient on GPUs.
Inspired by earlier work of Chen \& Chow and Nakatsukasa \& Freund, *Polar Express* adapts the update rule at each iteration by solving a minimax optimization problem.
We prove that this strategy minimizes error in a worst-case sense, allowing *Polar Express* to converge as rapidly as possible both in the early iterations and asymptotically.
We also address finite-precision issues, making it practical to use in `bfloat16`. When integrated into the Muon training framework, our method leads to consistent improvements in validation loss when training a GPT-2 model on one billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

TL;DR: We introduce a GPU-friendly algorithm for computing the polar decomposition of a matrix to low accuracy that is optimal in its class. This improves Muon.