On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer
This addresses the challenge of stable optimizer scaling for deep learning practitioners, offering a principled method for learning-rate transfer across model widths, though it builds incrementally on existing scaling theories.
The paper tackles the problem of designing neural optimizers that remain stable as network width increases by introducing mean-normalized operator norms, which enable width-independent smoothness bounds and practical optimizers like rescaled AdamW and row/column normalization. The result is MOGA, a width-aware optimizer that shows competitive performance with Muon and faster training in large-scale pre-training on GPT-2 and LLaMA.
A central question in modern deep learning is how to design optimizers whose behavior remains stable as the network width $w$ increases. We address this question by interpreting several widely used neural-network optimizers, including \textrm{AdamW} and \textrm{Muon}, as instances of steepest descent under matrix operator norms. This perspective links optimizer geometry with the Lipschitz structure of the network forward map, and enables width-independent control of both Lipschitz and smoothness constants. However, steepest-descent rules induced by standard $p \to q$ operator norms lack layerwise composability and therefore cannot provide width-independent bounds in deep architectures. We overcome this limitation by introducing a family of mean-normalized operator norms, denoted $\pmean \to \qmean$, that admit layerwise composability, yield width-independent smoothness bounds, and give rise to practical optimizers such as \emph{rescaled} \textrm{AdamW}, row normalization, and column normalization. The resulting learning rate width-aware scaling rules recover $μ$P scaling~\cite{yang2021tensor} as a special case and provide a principled mechanism for cross-width learning-rate transfer across a broad class of optimizers. We further show that \textrm{Muon} can suffer an $\mathcal{O}(\sqrt{w})$ worst-case growth in the smoothness constant, whereas a new family of row-normalized optimizers we propose achieves width-independent smoothness guarantees. Based on the observations, we propose MOGA (Matrix Operator Geometry Aware), a width-aware optimizer based only on row/column-wise normalization that enables stable learning-rate transfer across model widths. Large-scale pre-training on GPT-2 and LLaMA shows that MOGA, especially with row normalization, is competitive with Muon while being notably faster in large-token and low-loss regimes.