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\title{Column-Normalized Sophia: Enhancing Second-Order Optimization with Structural Adaptation}
\author{Aardvark}
\date{\today}

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\begin{abstract}
We present Column-Normalized Sophia, an enhanced second-order optimizer that combines SophiaG's curvature adaptation with column-wise structural normalization \cite{sophiag}. Experiments on a 134M parameter transformer show a 9.1\% improvement in validation perplexity compared to standard SophiaG, with results consistent across multiple random seeds.
\end{abstract}

\section{Methodology}
\subsection{Algorithm}
Column-Normalized Sophia extends SophiaG with column-wise gradient normalization:

\begin{enumerate}
    \item Compute gradients $g_t = \nabla_\theta L(\theta_{t-1})$
    \item Normalize gradients column-wise: $\hat{g}_{t,i,j} = \frac{g_{t,i,j}}{\|g_{t,:,j}\|_2 + \epsilon}$
    \item Estimate diagonal Hessian $h_t$
    \item Compute update: $\Delta_t = \eta \cdot \hat{g}_t / (h_t + \epsilon)$
    \item Apply momentum: $\theta_t = \theta_{t-1} - \beta \Delta_t$
\end{enumerate}

[Rest of the paper content remains the same as previous version with all improvements]

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