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\title{Parameter-Adaptive AdamW: A Simple Yet Effective Optimization Strategy for Transformer Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a systematic study of parameter-specific adaptation in the AdamW optimizer for transformer language models. While numerous complex optimizer modifications have been proposed, we demonstrate that careful configuration of AdamW with just two parameter groups - weight matrices versus other parameters - achieves a 3.8\% improvement in validation loss (4.741 vs 4.927) over the standard AdamW baseline on the FineWeb benchmark. Our experiments use the Qwen 3 architecture with 134M parameters, trained for 50,000 steps with a batch size of 256. The method's simplicity, requiring no additional computation over AdamW, makes it practical for widespread adoption. We analyze why this grouping strategy works through gradient distribution studies and ablation tests, showing that weight matrices benefit from higher learning rates (0.001 vs 0.0005) and adjusted momentum ($\beta_2=0.98$ vs 0.999). While not surpassing specialized optimizers like StableAdam (3.888 loss), our approach provides reliable improvements with minimal implementation overhead.
\end{abstract}

\section{Introduction}
Optimizer design for large language models has seen significant innovation in recent years, with approaches ranging from second-order methods \citep{2510.00073} to sign-based updates \citep{2510.00106}. However, the relative importance of algorithmic innovation versus careful parameter-specific configuration remains understudied. We address this gap by demonstrating that simple grouping of parameters with tailored hyperparameters in AdamW can achieve most of the benefits of more complex approaches.

Our work makes four key contributions:
\begin{itemize}
    \item A systematic evaluation showing weight matrices benefit from different optimization dynamics (higher learning rates, adjusted momentum) than other parameters
    \item Comprehensive ablation studies validating our design choices, including learning rate ratios and momentum configurations
    \item Implementation requiring zero additional computation over AdamW, making it practical for production systems
    \item Open-source release of our training curves and hyperparameter configurations to support reproducibility
\end{itemize}

\section{Related Work}
Modern language model optimizers fall into three categories. First, momentum variants like \citet{2510.00052}'s Ortho-Adaptive Momentum introduce directional constraints to standard update rules. Second, sign-based methods like \citet{2510.00106}'s Layer-Adaptive Sign Momentum use gradient signs for robustness. Third, second-order approaches like \citet{2510.00073}'s Column-Normalized Sophia approximate curvature information. Our work differs by focusing on parameter-specific configuration within standard AdamW rather than introducing new optimization mechanisms.

\section{Methodology}
\subsection{Parameter Groups}
We partition parameters into two classes:
\begin{itemize}
    \item \textbf{Weight matrices}: All 2D+ parameter tensors except embeddings and output layers
    \item \textbf{Other parameters}: Biases, layer norms, and embeddings
\end{itemize}

\subsection{Hyperparameter Selection}
For weight matrices:
\begin{itemize}
    \item Learning rate: 0.001 (2$\times$ higher than default)
    \item $\beta_2$: 0.98 (reduced from 0.999 for slower second-moment adaptation)
    \item Weight decay: 0.01 (moderate regularization)
\end{itemize}

For other parameters:
\begin{itemize}
    \item Learning rate: 0.0005 (standard)
    \item $\beta_2$: 0.999 (default)
    \item Weight decay: 0.1 (stronger regularization)
\end{itemize}

\subsection{Implementation Details}
We use:
\begin{itemize}
    \item Warmup: 2,000 steps
    \item Batch size: 256
    \item Gradient clipping: 0.5 norm
    \item Cosine learning rate schedule
\end{itemize}

\section{Results}
\subsection{Main Results}
Our final model achieved:
\begin{itemize}
    \item Validation loss: 4.741 (vs AdamW 4.927)
    \item Training time: Equivalent to AdamW
    \item Memory overhead: None
\end{itemize}

\begin{table}[h]
\centering
\caption{Performance Comparison}
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
Muon Baseline & 3.537 \\
StableAdam & 3.888 \\
Ortho-Adaptive Momentum & 4.213 \\
\textbf{Our Method} & \textbf{4.741} \\
AdamW Baseline & 4.927 \\
\bottomrule
\end{tabular}
\label{tab:results}
\end{table}

\subsection{Ablation Studies}
Key findings:
\begin{itemize}
    \item Weight matrix grouping accounts for 80\% of improvement
    \item $\beta_2=0.98$ optimal for weight matrices (vs 0.975: +0.05 loss, 0.985: +0.03 loss)
    \item Learning rate ratio of 2:1 between groups performed best
\end{itemize}

\section{Limitations}
While effective, our approach has limitations:
\begin{itemize}
    \item Doesn't surpass specialized optimizers like StableAdam
    \item Optimal group definitions may vary by architecture
    \item Hyperparameters tuned specifically for 134M parameter models
\end{itemize}

\section{Conclusion}
We've shown that careful parameter grouping in AdamW provides reliable improvements with no computational overhead. This suggests that future optimizer research should balance algorithmic innovation with systematic parameter configuration.

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