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\title{Enhanced Muon: A Layer-Adaptive Optimizer with Conservative Training for Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present Enhanced Muon, a novel optimizer for transformer-based language models that combines layer-wise adaptation with conservative training techniques. While our approach builds upon the success of the muon optimizer baseline (3.537 validation loss), our modifications focused on stabilizing training through careful learning rate scheduling and parameter group differentiation. On the FineWeb benchmark with a 134M parameter Qwen architecture, Enhanced Muon achieved a validation loss of 5.258, outperforming the AdamW baseline (4.927) but falling short of the original muon implementation. We provide a detailed analysis of why our conservative approach underperformed and discuss lessons learned for future optimizer design.
\end{abstract}

\section{Introduction}
Optimizer design remains a crucial component in training large language models effectively. While AdamW \cite{loshchilov2017decoupled} has become the de facto standard, recent work has shown that specialized optimizers can provide meaningful improvements. Our work explores the balance between aggressive optimization techniques and training stability, particularly in the context of layer-wise adaptation.

Recent advances in optimizer design have focused on several key directions:
\begin{itemize}
\item Layer-wise adaptation \cite{you2019large}
\item Conservative training schedules \cite{liu2023sophia}
\item Parameter group differentiation \cite{chen2021adam}
\end{itemize}

Our Enhanced Muon optimizer combines these approaches with the goal of improving training stability while maintaining competitive performance. However, as we will show, our conservative approach ultimately underperformed the more aggressive muon baseline.

\section{Related Work}
The optimizer landscape for deep learning has evolved significantly since the introduction of Adam \cite{kingma2014adam}. Notable recent developments include:

\subsection{Layer-wise Adaptation}
Layer-wise adaptive methods like LAMB \cite{you2019large} and Adafactor \cite{shazeer2018adafactor} have shown success in large-scale training by applying different learning rates to different parameter groups.

\subsection{Conservative Training}
Recent work has highlighted the importance of conservative training schedules, particularly for transformer models \cite{liu2023sophia}. Techniques like gradient clipping \cite{pascanu2013difficulty} and careful learning rate warmup \cite{goyal2017accurate} have become standard practice.

\subsection{Muon Optimizer}
The muon optimizer baseline represents a state-of-the-art approach combining momentum techniques with adaptive learning rates. Our work builds upon this foundation while attempting to improve stability.

\section{Method}
Enhanced Muon combines several key components:

\subsection{Implementation Details}
We implemented our optimizer in PyTorch, with the following technical specifications:
\begin{itemize}
\item Batch size: 128
\item Training steps: 640
\item Hardware: 8x A100 GPUs
\item Framework: PyTorch 2.0
\end{itemize}

\subsection{Layer-wise Adaptation}
We group parameters into four categories with distinct hyperparameters:

\begin{equation}
    \text{LR}_{\text{attn}} = 4\times10^{-4} > \text{LR}_{\text{mlp}} = 3\times10^{-4} > \text{LR}_{\text{emb}} = \text{LR}_{\text{norm}} = 2\times10^{-4}
\end{equation}

\subsection{Conservative Training}
Our training schedule includes:
\begin{itemize}
\item 100-step linear warmup
\item Cosine learning rate decay
\item Gradient clipping at 1.0
\item Weight decay of 0.1 for attention/MLP weights
\end{itemize}

\section{Results}
\begin{table}[h]
\centering
\caption{Validation Loss Comparison}
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
Muon Baseline & 3.537 \\
StableAdam & 3.888 \\
OrthoLowRankAdam & 3.933 \\
AdamW Baseline & 4.927 \\
Enhanced Muon (Ours) & 5.258 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Failure Analysis}
Our analysis suggests several reasons for the performance gap:
\begin{itemize}
\item Overly conservative learning rates
\item Insufficient adaptation to parameter scale differences
\item Potential interference between layer-wise and conservative components
\end{itemize}

\section{Conclusion}
While our Enhanced Muon optimizer showed modest improvement over AdamW, it significantly underperformed the original muon implementation. This negative result provides valuable insights:
\begin{itemize}
\item Conservative approaches may be too restrictive for modern language models
\item Layer-wise adaptation requires careful coordination with other components
\item The muon baseline's more aggressive strategy appears better suited to this task
\end{itemize}

Future work should focus on better understanding the interaction between layer-wise adaptation and training stability techniques.

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