Topic: polynomial update

David Persson presents Polar Express, a method for polar decomposition and the matrix sign function motivated by the Muon neural-network optimizer. It uses matrix multiplications suited to GPUs and adapts each polynomial update through a minimax problem to reduce worst-case error. The talk covers convergence, finite-precision implementation in bfloat16 and validation-loss improvements when training GPT-2 on FineWeb data. This is an in-person PACM IDeAS seminar at Princeton.

Robert Gower introduces Polar Express for the polar decomposition and matrix sign function, motivated by Muon neural-network training. Using only matrix-matrix products makes the method suited to high-throughput GPUs. Each iteration adapts its polynomial update through minimax optimization, building on Chen and Chow and Nakatsukasa and Freund. Worst-case error minimization gives rapid initial and asymptotic convergence. The talk addresses finite-precision implementation in bfloat16 and reports improved validation loss when training GPT-2 on one billion FineWeb tokens across several learning ra