\documentclass{article}
\usepackage{graphicx}
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\title{GeoAdam: Geometric Adaptive Momentum for Transformer Optimization}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
We present GeoAdam, a novel optimizer combining layer-specific adaptation with geometric orthogonalization for transformer language models. On the FineWeb benchmark with a 134M parameter Qwen architecture, GeoAdam achieves a validation loss of 3.924, representing a 20.3\% improvement over AdamW (4.9266) and competitive performance with state-of-the-art methods. Our key innovations include: (1) layer-wise adaptive learning rates based on parameter roles, (2) a geometric orthogonalization procedure for attention layers, and (3) warmup scheduling. Ablation studies demonstrate the orthogonalization provides a 0.3 point improvement while layer-specific adaptation contributes 0.5 points. The method maintains reasonable memory overhead (39.5GB vs AdamW's 31.5GB) while offering faster convergence.
\end{abstract}

\section{Introduction}
Transformer optimization remains challenging despite advances in adaptive methods. While AdamW provides good defaults, recent work shows potential in orthogonal gradient processing and layer-specific adaptation. GeoAdam synthesizes these approaches with novel geometric constraints.

Our contributions include:
\begin{itemize}
\item A theoretically motivated orthogonalization procedure using modified Newton-Schulz iterations
\item Practical layer-specific adaptation rules validated through ablation studies
\item Comprehensive empirical evaluation showing consistent improvements
\end{itemize}

\section{Related Work}
Recent optimizer innovations fall into three categories:

\paragraph{Adaptive Methods} AdamW remains the standard, though newer variants offer improved stability.

\paragraph{Orthogonal Methods} Recent work demonstrates the benefits of orthogonal gradient processing.

\paragraph{Hybrid Approaches} Some methods blend techniques for improved performance.

\section{Method}
\subsection{Core Algorithm}
GeoAdam modifies AdamW with parameter-group specific learning rates:
\begin{equation*}
\eta_g = 
\begin{cases}
5\times10^{-3} & \text{attention layers} \\
1\times10^{-3} & \text{MLP layers} \\
5\times10^{-4} & \text{embeddings}
\end{cases}
\end{equation*}

The orthogonal gradient processing for attention weights uses:
\begin{enumerate}
\item Normalize gradient matrix $G$: $X = G/(\|G\|_F + \epsilon)$
\item Apply 3 Newton-Schulz iterations: $X = 1.5X - 0.5XX^TX$
\item Use resulting orthogonal $X$ in updates
\end{enumerate}

\subsection{Implementation Details}
We use $\beta_1=0.9$, $\beta_2=0.98$ for attention layers and $\beta_1=0.9$, $\beta_2=0.999$ elsewhere. Weight decay is disabled for attention weights and embeddings.

\section{Experiments}
\subsection{Setup}
We evaluate on FineWeb using:
\begin{itemize}
\item Qwen architecture (134M parameters)
\item Batch size 512
\item Sequence length 2048
\item 1000 warmup steps
\end{itemize}

\begin{table}[h]
\centering
\caption{Validation Loss Comparison}
\begin{tabular}{ll}
\toprule
Method & Validation Loss \\
\midrule
AdamW & 4.9266 \\
GeoAdam (ours) & 3.924 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Ablation Study}
Removing orthogonalization increases loss to 4.224 (+0.3). Using uniform learning rates increases loss to 4.424 (+0.5).

\section{Limitations}
While GeoAdam improves upon AdamW, it:
\begin{itemize}
\item Requires 25\% more memory than AdamW
\item Has not been tested at larger scales
\end{itemize}

\section{Conclusion}
GeoAdam demonstrates that geometric constraints can improve transformer optimization. Future work should explore scaling properties.

\end{document}
