Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials

TL;DR

This paper introduces Chebyshev-optimized Newton-Schulz iteration (CANS) to improve matrix orthogonalization efficiency, optimizing coefficients for better performance in machine learning tasks like Riemannian optimization and the Muon optimizer.

Contribution

It develops a Chebyshev-based approach to optimize Newton-Schulz iteration coefficients, enhancing orthogonalization methods for deep learning applications.

Findings

Theoretical derivation of optimal coefficients for 3rd order Newton-Schulz iteration.

Application of Remez algorithm for higher-degree polynomial optimization.

Demonstrated improved efficiency in orthogonalization tasks in deep learning.

Abstract

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.