ARXIV:2603.09952 · OPTIMIZER RESEARCH · SUBMITTED 02 APR · 02:30 UTC · FRESHNESS STALE

VerifiedSource: PDF linkedPartialPaperPack: 3 of 4 citation fields filledMissingMissing fields: authorsPartialProof: unverified proof status

On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

arXiv

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Blocked on Code›Score3.0Evidence unverified

Opportunity summary

Pain MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Evidence 0 refs | 0 sources | 17% coverage

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PROBLEM

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths. We address this question by interpreting several widely used neural-network optimizers, including \textrm{AdamW} and \textrm{Muon}, as instances of steepest…

METHOD

Full abstract

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.

RESULT

ScienceToStartup currently rates this 3.0/10 on the public viability pass. 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.

WHY NOW

Optimizer Research moved forward this cycle; last verified April 2026. Public score 3.0/10.

Continue into Read for claims, analysis, references, and neighboring papers.

Opportunity summary

Score3.0

PainMOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Evidence0 refs | 0 sources | 17% coverage

Blockermissing authors

Analysis summary

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

VerifiedSource: PDF linkedPartialPaperPack: 3 of 4 citation fields filledMissingMissing fields: authorsPartialProof: unverified proof status

ARXIV:2603.09952 · OPTIMIZER RESEARCH · SUBMITTED 02 APR · 02:30 UTC · FRESHNESS STALE

VerifiedSource: PDF linkedPartialPaperPack: 3 of 4 citation fields filledMissingMissing fields: authorsPartialProof: unverified proof status

On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

arXiv

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Blocked on Code›Score3.0Evidence unverified

Opportunity summary

Pain MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Evidence 0 refs | 0 sources | 17% coverage

Blocker Evidence unverified

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PROBLEM

METHOD

Full abstract

RESULT

WHY NOW

Optimizer Research moved forward this cycle; last verified April 2026. Public score 3.0/10.

Continue into Read for claims, analysis, references, and neighboring papers.

Opportunity summary

Score3.0

PainMOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Evidence0 refs | 0 sources | 17% coverage

Blockermissing authors

Analysis summary

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

VerifiedSource: PDF linkedPartialPaperPack: 3 of 4 citation fields filledMissingMissing fields: authorsPartialProof: unverified proof status

Paper Pack

10.48550/arXiv.2603.09952

On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

Abstract

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PROBLEM

METHOD

RESULT

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Optimizer Research moved forward this cycle; last verified April 2026. Public score 3.0/10.

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Evidencepartial
MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths. 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.
Implicationpartial
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Verificationpartial
partial
Evidencepartial
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.
Implicationpartial
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
Verificationpartial
partial
Evidencepartial
ScienceToStartup currently rates this 3.0/10 on the public viability pass. 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.
Implicationpartial
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
Verificationpartial
partial
Evidencepartial
Optimizer Research moved forward this cycle; last verified April 2026. Public score 3.0/10.
Implicationpartial
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
Verificationpartial
partial

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MOGA introduces a new family of width-aware optimizers for stable learning-rate transfer across model widths.

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3.0/10 public viability

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References(58)

Why is Normalization Preferred? A Worst-Case Complexity Theory for Stochastically Preconditioned SGD under Heavy-Tailed Noise

2026Yuchen Fang, James Demmel et al.

$\mu$pscaling small models: Principled warm starts and hyperparameter transfer

2026Yuxin Ma, Nan Chen et al.

Preconditioning Benefits of Spectral Orthogonalization in Muon

2026Jianhao Ma, Yu Huang et al.

Understanding the Mechanisms of Fast Hyperparameter Transfer

2025Nikhil Ghosh, Denny Wu et al.

When do spectral gradient updates help in deep learning?

2025Damek Davis, D. Drusvyatskiy

MARS-M: When Variance Reduction Meets Matrices

2025Yifeng Liu, Angela Yuan et al.

Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methods

2025Andrey Veprikov, Arman Bolatov et al.

An Exploration of Non-Euclidean Gradient Descent: Muon and its Many Variants

2025M. Crawshaw, Chirag Modi et al.

Muon Outperforms Adam in Tail-End Associative Memory Learning

2025Shuche Wang, Fengzhuo Zhang et al.

Conda: Column-Normalized Adam for Training Large Language Models Faster

2025Junjie Wang, Pan Zhou et al.

A Minimalist Optimizer Design for LLM Pretraining

2025Athanasios Glentis, Jiaxiang Li et al.

Muon Optimizes Under Spectral Norm Constraints

2025Lizhang Chen, Jonathan Li et al.

GradPower: Powering Gradients for Faster Language Model Pre-Training

2025Mingze Wang, Jinbo Wang et al.

Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)

2025Artem Riabinin, Egor Shulgin et al.

ASGO: Adaptive Structured Gradient Optimization

2025Kang An, Yuxing Liu et al.

Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization

2025Dmitry Kovalev

Structured Preconditioners in Adaptive Optimization: A Unified Analysis

2025Shuo Xie, Tianhao Wang et al.

Muon is Scalable for LLM Training

2025Jingyuan Liu, Jianling Su et al.

Training Deep Learning Models with Norm-Constrained LMOs

2025T. Pethick, Wanyun Xie et al.

Gradient Multi-Normalization for Stateless and Scalable LLM Training

2025Meyer Scetbon, Chao Ma et al.

Showing 20 of 58 references

CITED BY

No citing papers are indexed in the public S2S graph yet. This is an explicit zero-signal state, not a hidden lookup.

Foundation

Prior WorkDoes Your Optimizer Care How You Normalize? Normalization-Optimizer Coupling in LLM Training

3.0

Extension

Builds On ThisMuon Dynamics as a Spectral Wasserstein Flow

0.0

Builds On ThisDeriving Hyperparameter Scaling Laws via Modern Optimization Theory

2.0

Builds On ThisHow the Optimizer Shapes Learned Solutions in Equivariant Neural Networks

2.0

Commercially relevant

Higher ViabilityRethinking Language Model Scaling under Transferable Hypersphere Optimization

7.0

Higher ViabilityMuonEq: Balancing Before Orthogonalization with Lightweight Equilibration

4.0

Higher ViabilityMano: Restriking Manifold Optimization for LLM Training

5.0

Higher Viability$μ$pscaling small models: Principled warm starts and hyperparameter transfer

4.0

Higher ViabilityMuon is Not That Special: Random or Inverted Spectra Work Just as Well

7.0

Conflicting

Competing ApproachOptimizer-Induced Mode Connectivity: From AdamW to Muon

3.0

Owned Distribution

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Paper: 2603.09952
Route: /paper/on-the-width-scaling-of-neural-optimizers-under-matrix-operator-norms-i-row-column-normalization-and-hyperparameter-tran
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Schema validation

paper-r2 contract: present
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OpenAPI path parity: /api/openapi.json
MCP resource parity: paper-workspace

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Page Freshness

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Page Freshness

Paper proof surface

Canonical route: /paper/on-the-width-scaling-of-neural-optimizers-under-matrix-operator-norms-i-row-column-normalization-and-hyperparameter-tran

stale

Proof freshness: stale
Proof status: unverified
Display score: 3/10
Last proof check: 2026-04-02
Score updated: 2026-04-02
Score fresh until: 2026-05-02
References: 0
Source count: 0
Coverage: 17%

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On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

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MCP example

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Buildability Receipt

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Not build-ready: On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

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Ignoreblocked

Subject: On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

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Evidence ids

Receipt path

/buildability/on-the-width-scaling-of-neural-optimizers-under-matrix-operator-norms-i-row-column-normalization-and-hyperparameter-tran

Paper ref

on-the-width-scaling-of-neural-optimizers-under-matrix-operator-norms-i-row-column-normalization-and-hyperparameter-tran

arXiv id

2603.09952

Freshness

Generated at

2026-04-02T02:30:40.136Z

Evidence freshness

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Last verification

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On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer

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