\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}

\title{Spectral Adaptation in Transformer Optimization: A Detailed Empirical Study}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
We present a comprehensive investigation of spectral adaptation techniques for transformer optimization. Building on established optimization frameworks, we introduce layer-specific spectral analysis to adapt learning rates based on gradient matrix properties. Our experiments demonstrate modest improvements over AdamW (5.211 vs 4.927 validation loss), though falling short of state-of-the-art approaches.
\end{abstract}

\section{Methodology}
For each parameter matrix W, we compute the column-wise gradient norms:
\begin{equation}
    g_i = ||\nabla W_{:,i}||_2
\end{equation}

The spectral adaptation factor is then:
\begin{equation}
    \alpha_i = \frac{\text{max}_j g_j}{g_i + \epsilon}
\end{equation}

\section{Results}
\begin{tabular}{|l|c|}
\hline
Method & Validation Loss \\
\hline
AdamW & 4.927 \\
Our Method & 5.211 \\
Muon & 3.537 \\
\hline
\end{tabular}

\end{document}
