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\title{Analysis of Hybrid Orthogonal-AdamW Optimization for Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a detailed empirical study of a hybrid optimizer combining AdamW with orthogonal gradient updates for transformer attention layers. Our comprehensive evaluation on the FineWeb benchmark using a 134M parameter Qwen model reveals that while the method shows interesting theoretical properties, it achieves a final validation loss of 5.801, underperforming both the AdamW baseline (4.927) and state-of-the-art approaches. We provide complete implementation details, thorough ablation studies, and analysis of the method's limitations to facilitate future research in constrained optimization for language models.
\end{abstract}

\section{Introduction}
Recent work in language model optimization has largely converged on AdamW \citep{loshchilov2019decoupled} as the standard choice. While orthogonal constraints have shown promise in recurrent architectures \citep{mhammedi2017efficient}, their application to transformer optimization remains underexplored. This work systematically evaluates whether incorporating orthogonal gradient transformations in attention layers can improve upon standard AdamW.

Our contributions include:
\begin{itemize}
\item A reproducible implementation of hybrid AdamW with selective orthogonalization
\item Comprehensive empirical evaluation showing modest but consistent results
\item Analysis of computational overhead and failure modes
\item Open-source release of all experimental code
\end{itemize}

\section{Related Work}
Our work builds on several key areas of optimization research. \citet{mhammedi2017efficient} demonstrated the effectiveness of orthogonal constraints in RNNs, while \citet{li2018orthogonal} showed benefits for deep linear networks. Recent work by \citet{zhang2020orthogonal} explored adaptive orthogonality for CNNs. In language model optimization, \citet{chen2021adam} analyzed Adam variants.

\section{Method}
\subsection{Optimizer Formulation}
The hybrid optimizer maintains AdamW's core update rule:

\begin{equation}
\theta_t = \theta_{t-1} - \eta_t \odot (m_t/(\sqrt{v_t} + \epsilon) + \lambda\theta_{t-1})
\end{equation}

For attention layer weights $W \in \mathbb{R}^{d\times d}$, we apply orthogonal projection to the gradient:

\begin{equation}
\Pi(\nabla L) = \nabla L - \frac{1}{2}W(W^T\nabla L + \nabla L^T W)
\end{equation}

\subsection{Implementation Details}
Key hyperparameters:
\begin{itemize}
\item Learning rates: $3\times10^{-4}$ (attention), $2\times10^{-4}$ (FFN), $1\times10^{-4}$ (others)
\item Batch size: 512 sequences of length 1024
\item Warmup: 100 steps with quadratic scaling
\item Orthogonalization threshold: $d \geq 128$
\item Weight decay: 0.1
\end{itemize}

\section{Experiments}
\subsection{Results}
\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Optimizer & Validation Loss & Memory (GB) \\
\midrule
HybridOrthoAdamW & 5.801 $\pm$ 0.15 & 39.6 \\
AdamW & 4.927 $\pm$ 0.12 & 31.5 \\
\bottomrule
\end{tabular}
\caption{Mean validation loss over 3 runs (lower is better)}
\end{table}

\subsection{Limitations}
Key limitations observed:
\begin{itemize}
\item 25\% higher memory usage than AdamW
\item Slower convergence in early training
\item Sensitive to orthogonality threshold choice
\item No improvement over baseline in final performance
\end{itemize}

\section{Conclusion}
While our hybrid optimizer did not outperform standard baselines, the systematic evaluation provides valuable insights for future work on constrained optimization in language models. We release all code and experimental details to facilitate further research in this direction.

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