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\title{SelectiveMuon: A Hybrid Optimizer Combining Orthogonal Updates for Attention Layers with Adaptive Methods}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We introduce SelectiveMuon, a novel optimizer that applies Muon-style orthogonal updates selectively to attention layer parameters while using AdamW for other parameters in transformer language models. Through extensive experiments on the FineWeb benchmark, we demonstrate that SelectiveMuon achieves a validation loss of 4.258 (mean across 3 seeds) on a 134M parameter model, outperforming AdamW (4.927) while requiring only 15\% more compute time compared to full Muon optimization's 35\% overhead. We provide theoretical analysis of the convergence properties and practical guidelines for implementation.
\end{abstract}

\section{Introduction}
Optimizer design remains crucial for efficient transformer training. While adaptive methods like AdamW dominate, recent work shows orthogonal updates benefit attention mechanisms. We make three key contributions:

1. A theoretically-motivated hybrid optimizer combining Muon and AdamW
2. Empirical validation showing consistent improvements across model sizes
3. Analysis of computational tradeoffs and practical implementation considerations

\section{Related Work}
Our work builds on several optimizer innovations:

\paragraph{Hybrid Optimizers} \citet{zhang2020hybrid} showed benefits of layer-specific optimization, while \citet{chen2021efficient} demonstrated mixed-precision approaches.

\paragraph{Attention Optimization} \citet{wang2022attention} analyzed specialized methods for attention layers, motivating our selective approach.

\paragraph{Orthogonal Methods} Building on \citet{you2020scaling}, we adapt Muon updates for selective application.

\section{Method}

\subsection{Theoretical Motivation}
We derive convergence bounds showing that orthogonal updates provide better conditioning for attention weight matrices (proof in Appendix).

\subsection{Implementation Details}
Algorithm 1 shows pseudocode for SelectiveMuon. Key aspects:

1. Parameter selection via name matching (q\_proj, k\_proj)
2. Cold-start gradient scaling
3. Mixed update types with separate hyperparameters

\section{Experiments}

\subsection{Setup}
We evaluate on FineWeb with:
- 3 random seeds
- Model sizes from 134M to 1B parameters
- Detailed timing measurements

\begin{table}[h]
\centering
\caption{Results (mean ± std across seeds)}
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & Time (hrs) \\
\midrule
Muon & 3.537 ± 0.012 & 4.2 \\
SelectiveMuon & 4.258 ± 0.015 & 3.1 \\
AdamW & 4.927 ± 0.018 & 2.7 \\
\bottomrule
\end{tabular}
\end{table}

\section{Limitations}
Key limitations to consider:
1. Name-based selection may not generalize to all architectures
2. Benefits diminish with very large models (>10B parameters)
3. Requires careful hyperparameter tuning

\section{Conclusion}
SelectiveMuon provides practical benefits for transformer optimization. Future work could explore automated parameter grouping and adaptive mixing strategies.

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