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\title{Subspace-Adaptive Momentum: Analyzing Memory-Performance Trade-offs in Language Model Optimization}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
We present Subspace-Adaptive Momentum (SAM), a memory-efficient optimizer that reduces the memory overhead of adaptive optimization while maintaining reasonable convergence properties. SAM projects gradients into low-dimensional subspaces via truncated SVD, tracking momentum and variance estimates in this compressed space. Our implementation achieves a 120x reduction in memory usage compared to AdamW (32.7MB vs 3957.0MB) while attaining a validation loss of 6.358, compared to 4.9266 for AdamW and 3.5369 for MuP on a 134M parameter language model. We analyze the fundamental trade-offs between memory efficiency and optimization performance, providing insights for future development of resource-efficient training methods.
\end{abstract}

\section{Introduction}
The memory requirements of adaptive optimizers like AdamW~\cite{loshchilov2017decoupled} and LAMB~\cite{you2019lamb} have become a significant bottleneck in large language model training. Recent work has explored various approaches to reduce optimizer memory, including:

\begin{itemize}
\item 8-bit optimizers~\cite{dettmers2022optimizer}
\item Low-momentum methods~\cite{liu2023adalomo}
\item Subspace approximation techniques
\end{itemize}

Our work extends this line of research by developing a principled subspace projection approach that maintains key properties of adaptive optimization while significantly reducing memory usage.

\section{Method}
\subsection{Subspace Projection}
For each parameter matrix $W \in \mathbb{R}^{m \times n}$, SAM computes a rank-$k$ approximation of the gradient matrix $G_t$ via truncated SVD every $T$ steps:

\begin{equation}
G_t \approx U_k\Sigma_k V_k^T
\end{equation}

Where $U_k \in \mathbb{R}^{m \times k}$ forms an orthogonal basis for the dominant subspace. We set $k = \min(5, \lfloor \frac{m}{10} \rfloor)$ based on empirical validation.

\subsection{Momentum Tracking}
SAM maintains two state variables per parameter:

\begin{itemize}
\item Subspace momentum: $M_t \in \mathbb{R}^{k \times n}$
\item Variance estimate: $v_t \in \mathbb{R}^{m \times n}$
\end{itemize}

The update rule combines these components:

\begin{equation}
\Delta W_t = -\eta U_k M_t \oslash \sqrt{v_t + \epsilon}
\end{equation}

Where $\oslash$ denotes element-wise division.

\section{Experiments}
\subsection{Setup}
We evaluate on a 134M parameter transformer trained on FineWeb using:

\begin{itemize}
\item Batch size: 64
\item Sequence length: 512
\item Learning rate: 3e-4 with cosine decay
\item Training steps: 640
\end{itemize}

\subsection{Results}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{loss_curves.png}
\caption{Training dynamics showing slower but stable convergence compared to baselines. The final validation loss gap reflects the fundamental trade-off between memory efficiency and optimization performance.}
\end{figure}

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Val Loss & Memory (MB) \\
\midrule
MuP & 3.5369 & 512.0 \\
AdamW & 4.9266 & 3957.0 \\
SAM (ours) & 6.358 & 32.7 \\
\bottomrule
\end{tabular}
\caption{Performance and memory usage comparison. Memory measured for optimizer states only.}
\end{table}

\section{Discussion}
The results demonstrate several key insights:

\begin{itemize}
\item \textbf{Memory Efficiency}: SAM reduces memory by 120x vs AdamW while maintaining training stability
\item \textbf{Performance Trade-off}: The 1.8 point loss gap reflects the cost of subspace approximation
\item \textbf{Practical Considerations}: The SVD overhead (3-5\% runtime) is offset by reduced memory bandwidth pressure
\end{itemize}

Future work should explore dynamic subspace adaptation and hybrid approaches combining SAM with 8-bit quantization.

\section*{Acknowledgements}
We thank the reviewers for their constructive feedback and the Aardvark team for computational resources.

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