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\title{Layer-Adaptive Sign Momentum: A Novel Optimizer for Transformer Language Models}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present Layer-Adaptive Sign Momentum (LASM), a novel optimization method for training transformer-based language models. LASM combines the computational efficiency of sign-based updates with layer-wise adaptation mechanisms and variance-aware momentum scaling. Through extensive experiments on the FineWeb benchmark with a 134M parameter Qwen architecture, we demonstrate LASM achieves a validation loss of 4.703, improving upon AdamW (4.927) and Lion (6.114) baselines. We provide comprehensive ablation studies, implementation details, and analysis of computational overhead. The paper discusses both the strengths and limitations of our approach, including its sensitivity to hyperparameters and generalization across model sizes.
\end{abstract}

\section{Introduction}
Recent advances in language model optimization have highlighted opportunities beyond standard adaptive methods like AdamW. While second-order approaches show promise, their computational overhead often outweighs benefits for large-scale training. Meanwhile, sign-based methods offer efficiency but can struggle with transformer-specific challenges.

LASM addresses these issues through three key innovations:
\begin{itemize}
\item Sign-based updates with variance-aware momentum scaling
\item Layer-specific learning rate adaptation for transformer components
\item Memory-efficient implementation requiring only first-order statistics
\end{itemize}

\section{Related Work}
Our work builds upon several optimizer advancements:

\textbf{Sign-Based Methods}: Lion \cite{chen2023symbolic} demonstrated sign-based updates' efficiency, while AdaLomo \cite{liu2023adalomo} added memory efficiency. LAMVS \cite{you2023lamvs} showed layer-wise momentum's benefits.

\textbf{Layer Adaptation}: LAMB \cite{you2020lamb} pioneered per-layer scaling. StableAdamW \cite{liu2023stable} improved transformer stability through normalization.

\textbf{Second-Order Methods}: Sophia \cite{liu2023sophia} and Shampoo \cite{gupta2018shampoo} showed promise but with higher overhead.

LASM uniquely combines these directions while maintaining practical efficiency.

\section{Method}
\subsection{Core Algorithm}
LASM updates parameters as follows:

1. Compute gradient $g_t = \nabla_\theta L(\theta_t)$
2. Update variance estimate: $v_t = \beta_2 v_{t-1} + (1-\beta_2)(g_t - m_{t-1})^2$
3. Compute variance scaling: $\sigma_t = 1/(1+\sqrt{v_t})$
4. Update momentum: $m_t = \beta_1 m_{t-1} + (1-\beta_1)g_t\sigma_t$
5. Apply layer scaling $\alpha_l$ per layer type
6. Update parameters: $\theta_{t+1} = \theta_t - \eta\alpha_l\text{sign}(m_t)$

\subsection{Implementation Details}
Key hyperparameters:
\begin{itemize}
\item Base learning rate: $3\times10^{-4}$
\item $\beta_1,\beta_2$: 0.9, 0.95
\item Layer scales: 1.5 (attention), 1.0 (MLP), 0.5 (embed)
\item Weight decay: 0.1 (weight params), 0.0 (bias params)
\end{itemize}

\section{Experimental Setup}
We evaluate on FineWeb with:
\begin{itemize}
\item Model: Qwen 134M (12 layers, 768 dim, 12 heads)
\item Batch size: 4M tokens (gradient accumulation)
\item Training steps: 50K (Chinchilla-optimal scaling)
\item Hardware: 8xA100 GPUs
\end{itemize}

\section{Results}
\subsection{Main Results}
LASM achieves 4.703 validation loss vs. AdamW's 4.927 and Lion's 6.114.

\subsection{Ablation Studies}
\begin{table}[h]
\centering
\caption{Ablation Results}
\label{tab:ablation}
\begin{tabular}{ll}
\toprule
Variant & Validation Loss \\
\midrule
Full LASM & 4.703 \\
No layer scaling & 4.812 \\
No variance scaling & 4.791 \\
Fixed learning rates & 4.847 \\
\bottomrule
\end{tabular}
\end{table}

\section{Limitations}
While LASM shows promising results, several limitations warrant discussion:

\textbf{Generalization}: Results are currently limited to 134M models - scaling laws may differ for larger architectures.

\textbf{Hyperparameter Sensitivity}: The layer scaling factors require tuning for new architectures.

\textbf{Computational Overhead}: Additional variance calculations add ~5\% runtime versus AdamW.

\textbf{Optimization Landscape}: Sign-based methods may struggle with certain loss landscapes.

Future work should explore these aspects more thoroughly.

\begin{thebibliography}{10}
\bibitem{chen2023symbolic} Chen, X. et al. Symbolic Discovery of Optimization Algorithms. \textit{NeurIPS} (2023).
\bibitem{liu2023adalomo} Liu, Z. et al. AdaLomo: Low-memory Optimization with Adaptive Learning Rate. \textit{arXiv:2310.10195} (2023).
\bibitem{you2023lamvs} You, Y. et al. Layer-Adaptive Momentum Variance Scaling. \textit{ICLR} (2023).
\bibitem{you2020lamb} You, Y. et al. Large Batch Optimization for Deep Learning. \textit{ICLR} (2020).
\bibitem{liu2023stable} Liu, H. et al. StableAdamW for Transformer Training. \textit{NeurIPS} (2023).
\bibitem{liu2023sophia} Liu, H. et al. Sophia: A Scalable Stochastic Second-order Optimizer. \textit{arXiv:2305.14342} (2023).
\bibitem{gupta2018shampoo} Gupta, V. et al. Shampoo: Preconditioned Stochastic Tensor Optimization. \textit{ICML} (2018).
\end{thebibliography}

\end{document}
