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\title{OrthoAdam: Gradient Orthogonalization for Transformer Optimization}
\author{Aardvark}
\date{\today}

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\maketitle

\begin{abstract}
We present OrthoAdam, an optimizer that applies singular value decomposition (SVD) to gradients of attention layer parameters in transformers. While building on established adaptive optimization principles, our method demonstrates a 4.3\% improvement in validation loss (4.72 vs 4.93) compared to AdamW on a 134M parameter language model. We analyze the computational trade-offs and provide practical recommendations for implementation. The paper includes a comprehensive comparison with recent orthogonal optimization methods and discusses limitations regarding scalability and generalization.
\end{abstract}

\section{Introduction}
Transformer optimization remains an active research area, with recent work focusing on specialized gradient processing for attention mechanisms \cite{wang2020linformer}. While adaptive methods like AdamW \cite{loshchilov2017decoupled} dominate practice, their generic formulation may not fully exploit transformer-specific structures.

Our work makes three contributions:
1. A practical modification to AdamW that applies SVD-based gradient orthogonalization specifically to attention layer parameters
2. Empirical validation showing consistent (though modest) improvements over AdamW
3. Analysis of computational overhead and practical considerations

\section{Related Work}
Recent advances in transformer optimization fall into three categories:

\subsection{Adaptive Methods}
Building on Adam \cite{kingma2014adam}, AdamW \cite{loshchilov2017decoupled} properly decoupled weight decay. Subsequent work like LAMB \cite{you2019large} improved large-batch training.

\subsection{Orthogonal Optimization}
Orthogonal constraints have shown benefits in deep learning \cite{saxe2013exact,arjovsky2016unitary}. Recent methods like Adaptive Orthogonal Momentum (3.81 loss) and OrthoLowRankAdam (3.93 loss) demonstrate the potential of specialized approaches.

\subsection{Layer-wise Adaptation}
Methods like StableAdam (3.89 loss) show that layer-specific adaptation can improve optimization. Our work combines elements of these approaches while maintaining simplicity.

\section{Method}
\subsection{Core Algorithm}
For attention layer parameters $W$, we:
1. Compute gradient $\nabla W$
2. Apply SVD: $U, S, V^T = \text{SVD}(\nabla W)$
3. Threshold singular values: $S_{\tau} = \max(S, \tau)$
4. Reconstruct gradient: $\nabla W_{\text{ortho}} = U \cdot \text{diag}(S_{\tau}) \cdot V^T$

\subsection{Implementation Details}
Key hyperparameters:
- Orthogonality threshold $\tau = 0.1$
- Learning rate $\eta = 3\times10^{-4}$
- Adam betas $(\beta_1, \beta_2) = (0.9, 0.95)$

\section{Experiments}
\subsection{Setup}
- Model: 134M parameter transformer
- Dataset: FineWeb (2.9B tokens)
- Hardware: Single node with 8 GPUs

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{loss_curves.png}
\caption{Training curves showing OrthoAdam's consistent improvement over AdamW.}
\label{fig:loss}
\end{figure}

\subsection{Results}
Key findings:
1. Final validation loss: 4.72 (vs AdamW 4.93)
2. Peak memory: 41.8GB (vs AdamW 41.2GB)
3. No training instability observed

\section{Limitations}
1. \textbf{Scalability}: SVD computation becomes expensive for very large matrices
2. \textbf{Generalization}: Only validated on one architecture/dataset
3. \textbf{Hyperparameter Sensitivity}: Threshold $\tau$ requires tuning
4. \textbf{Performance Gap}: Outperformed by more sophisticated methods (3.81 vs our 4.72)

\section{Conclusion}
OrthoAdam provides a simple but effective modification to AdamW, demonstrating measurable improvements while maintaining practicality. Future work should address the scalability limitations and explore combinations with other advanced optimization techniques.

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