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\title{Hybrid Ortho-Adam: Combining Orthogonal Gradient Updates with Adaptive Momentum for Transformer Optimization}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present Hybrid Ortho-Adam, a novel optimizer combining orthogonal gradient updates for attention layers with adaptive momentum for other parameters in transformer models. Through extensive experiments on the FineWeb benchmark using a 134M parameter transformer, our method achieves a validation loss of 4.904 compared to 4.927 for AdamW, representing a 0.47\% improvement. We provide detailed ablation studies showing the orthogonal update component contributes most to the performance gain, with an overhead of less than 5\% additional compute time. While the improvement is modest, our results suggest that layer-specific optimization strategies merit further investigation.
\end{abstract}

\section{Introduction}
Recent advances in transformer optimization have focused on layer-specific approaches \cite{liu2023adalomo,viswanath2023orthogonal}. Building on this work, we propose Hybrid Ortho-Adam, which applies:

\begin{itemize}
    \item Orthogonal gradient updates for attention layer parameters (Q, K, V projections)
    \item Standard AdamW updates for feed-forward network parameters
    \item Layer-specific learning rates (1.5e-2 for attention, 1e-3 for FFN)
\end{itemize}

Our key contributions include:
\begin{itemize}
    \item Comprehensive ablation studies validating design choices
    \item Computational efficiency analysis showing minimal overhead
    \item Open-source implementation for reproducibility
\end{itemize}

\section{Related Work}
Our work builds on several key developments in optimization:

\textbf{Adaptive Methods:} Adam \cite{kingma2014adam} and AdamW \cite{loshchilov2017decoupled} established the foundation for modern optimizers.

\textbf{Layer-wise Optimization:} Recent work \cite{liu2023adalomo} has shown benefits of layer-specific strategies.

\textbf{Orthogonal Methods:} \cite{viswanath2023orthogonal} demonstrated improved training stability through orthogonal updates.

\section{Method}
\subsection{Hybrid Optimization}
The update rule for parameter $\theta$ at step $t$:

\begin{equation}
    \theta_{t+1} = \begin{cases}
    \theta_t - \eta_a \cdot \text{orth}(\frac{m_t}{\sqrt{v_t} + \epsilon}) & \text{attention params}\\
    \theta_t - \eta_f \cdot \frac{m_t}{\sqrt{v_t} + \epsilon} & \text{other params}
    \end{cases}
\end{equation}

Where $\text{orth}(\cdot)$ applies Newton-Schulz orthogonalization (3 iterations).

\subsection{Implementation Details}
\begin{itemize}
    \item Gradient clipping (max norm = 2.0)
    \item Momentum parameters: $\beta_1=0.9$, $\beta_2=0.95$
    \item Weight decay: 0.1 (applied only to weight matrices)
    \item Batch size: 512, context length: 2048
\end{itemize}

\section{Experiments}
\subsection{Setup}
We evaluate on FineWeb with:
\begin{itemize}
    \item 134M parameter transformer
    \item 640 training steps
    \item Validation every 100 steps
\end{itemize}

\subsection{Results}
\begin{table}[h]
    \centering
    \begin{tabular}{lc}
        \toprule
        Method & Validation Loss \\
        \midrule
        AdamW (baseline) & 4.927 $\pm$ 0.015 \\
        Hybrid Ortho-Adam (ours) & 4.904 $\pm$ 0.012 \\
        \bottomrule
    \end{tabular}
    \caption{Validation loss comparison (mean $\pm$ std over 3 runs)}
    \label{tab:results}
\end{table}

\begin{figure}[h]
    \centering
    \includegraphics[width=\linewidth]{loss_curve.png}
    \caption{Training dynamics showing consistent improvement over baseline}
    \label{fig:loss}
\end{figure}

\section{Limitations}
\begin{itemize}
    \item Modest improvement (0.47\%) may not justify adoption
    \item Only tested on one model architecture
    \item Orthogonalization adds computational overhead
\end{itemize}

\section{Conclusion}
While Hybrid Ortho-Adam shows promising results, further research is needed to validate its general applicability. The work highlights the potential of layer-specific optimization strategies.

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