AuON: A Linear-time Alternative to Orthogonal Momentum Updates

Orthogonal momentum gradient updates have emerged to overcome the limitations of vector-based optimizers like Adam. The vector-based optimizer Adam suffers from high memory costs and ill-conditioned momentum gradient updates. However, traditional Orthogonal momentum approaches, such as SVD/QR decomposition, suffer from high computational and memory costs and underperform compared to well-tuned SGD with momentum. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and approximate orthogonal matrices via Newton-Schulz iterations, which gives better GPU utilization, active high TFLOPS, and reduces memory usage by up to 3x. Nevertheless, Muon(Vanilla) suffers from exploding attention logits and has cubic computation complexity. In this paper, we deep dive into orthogonal momentum gradient updates to find the main properties that help Muon achieve remarkable performance. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without approximate orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. AuON has an automatic "emergency brake" to handle exploding attention logits. We further introduce a hybrid variant, Hybrid-AuON, that applies the linear transformations with Newton-Schulz iterations, which outperforms Muon in the language modeling tasks. Code is available at: https://github.com/ryyzn9/AuON