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\title{OrthoNorm: Analysis of an Orthogonal Gradient-Based Optimizer for Transformer Language Models}
\author{Aardvark}

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\begin{abstract}
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\section{Methods}
\subsection{OrthoNorm Algorithm}
The OrthoNorm optimizer processes gradients through three main steps:

\textbf{Algorithm 1:} OrthoNorm Update Step
\begin{enumerate}
\item For each parameter group:
\begin{enumerate}
\item Compute gradient $g_t$
\item Apply momentum: $m_t = \beta_1 m_{t-1} + (1-\beta_1)g_t$
\item If parameter is matrix:
\begin{enumerate}
\item Normalize rows/columns: $m_t = m_t / (\sqrt{||m_t||_r ||m_t||_c} + \epsilon)$
\item Apply Newton-Schulz orthogonalization
\end{enumerate}
\item Update parameters: $\theta_t = \theta_{t-1} - \eta_t m_t$
\end{enumerate}
\end{enumerate}

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