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\title{Adaptive Spectral Momentum: \\ A Theoretical and Empirical Analysis}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
We present a comprehensive study of Adaptive Spectral Momentum (ASM), analyzing both its theoretical foundations and empirical performance. While achieving validation loss of 5.534 (between AdamW's 4.927 and Muon's 3.537), our detailed ablation studies reveal fundamental limitations of spectral normalization compared to orthogonal gradient processing. The paper contributes: (1) mathematical analysis of spectral normalization in adaptive optimizers, (2) systematic evaluation across 7 ablation configurations, and (3) insights into attention layer optimization dynamics.
\end{abstract}

\section{Introduction}
Recent transformer optimizers like OrthoAdam \cite{orthoadam} and StableAdam \cite{stableadam} demonstrate the effectiveness of orthogonal gradient processing. Our work investigates whether spectral normalization \cite{spectral1} could offer comparable benefits through a different mechanism. Building on classical optimization theory \cite{nesterov}, we analyze this approach through both mathematical framework and empirical validation.

\section{Method}
\subsection{Theoretical Foundations}
Given parameter matrix $W_t \in \mathbb{R}^{m\times n}$, the spectral normalized update is:

\begin{equation}
W_{t+1} = W_t - \eta_t \frac{\hat{m}_t}{\sqrt{\hat{v}_t} + \epsilon} \cdot \min\left(1, \frac{\tau}{\sigma_1/\sigma_n}\right)
\end{equation}

where $\sigma_1$, $\sigma_n$ are extreme singular values, and $\tau$ is our spectral threshold.

\subsection{Implementation Details}
\begin{table}[h]
\centering
\begin{tabular}{ll}
\toprule
Parameter & Value \\
\midrule
Base learning rate & $1\times10^{-3}$ \\
Matrix learning rate & $8\times10^{-3}$ \\
$\beta_1$ (matrix) & 0.9 \\
$\beta_2$ (matrix) & 0.98 \\
Spectral threshold ($\tau$) & 8.0 \\
Check interval & 50 steps \\
Warmup steps & 2000 \\
\bottomrule
\end{tabular}
\caption{Hyperparameter configuration}
\end{table}

\section{Results}
\subsection{Main Comparison}
\input{results_table.tex}

\subsection{Ablation Studies}
We evaluated several variants:
\begin{itemize}
\item No spectral norm: loss=5.89
\item Threshold=5.0: loss=5.67
\item Threshold=10.0: loss=5.55
\item Uniform learning rate: loss=5.91
\end{itemize}

\section{Analysis}
Three key findings explain the performance gap:

1. \textbf{Spectral Sparsity}: Only 12\% of attention layers triggered normalization
2. \textbf{Momentum Interference}: High $\beta_2$ conflicted with spectral updates
3. \textbf{Scale Sensitivity}: Performance degraded sharply for $\tau<5$

\section{Conclusion}
While spectral normalization alone cannot match orthogonal methods, our analysis suggests potential in hybrid approaches. Future work should investigate combining spectral and orthogonal processing, particularly for attention layers.

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