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\title{SignCurv: Combining Sign-Based Updates with Adaptive Curvature for Transformer Optimization}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present SignCurv, a novel optimizer combining sign-based gradient updates with lightweight curvature adaptation for transformer language models. Our method addresses key limitations in existing optimizers by (1) using sign-based updates for stable optimization across different parameter scales, (2) incorporating adaptive curvature information through diagonal Hessian approximations, and (3) implementing architecture-aware learning rate scheduling. Experiments on the FineWeb dataset demonstrate SignCurv achieves competitive performance (validation loss 4.018) while maintaining training stability. Compared to AdamW (loss 4.927), our method shows a 18.4\% relative improvement, though it does not surpass state-of-the-art methods like Muon (3.537).
\end{abstract}

\section{Introduction}
Recent advances in language model optimization have primarily focused on adaptive momentum methods \cite{adam,adamw,muon}. While effective, these approaches often struggle with training stability and parameter scale sensitivity. Our work revisits sign-based optimization \cite{signsgd,velo} through the lens of modern transformer architectures, combining it with adaptive curvature information from recent second-order methods \cite{shampoo,symbolic}.

\subsection{Key Contributions}
\begin{itemize}
\item Novel optimizer combining sign-based updates with diagonal Hessian approximations
\item Architecture-aware learning rate scheduling with warmup and cosine decay
\item Comprehensive empirical evaluation showing competitive performance
\item Detailed ablation studies and hyperparameter sensitivity analysis
\end{itemize}

\section{Related Work}
Our work builds upon several key developments in optimization:

\subsection{Adaptive Methods}
Adam \cite{adam} and AdamW \cite{adamw} demonstrated the effectiveness of adaptive momentum, while recent work like Muon \cite{muon} and VeLO \cite{velo} have pushed state-of-the-art performance.

\subsection{Sign-Based Methods}
SignSGD \cite{signsgd} showed promise for large-scale distributed training, with subsequent improvements in \cite{practical}.

\subsection{Second-Order Methods}
Shampoo \cite{shampoo} and Symbolic Discovery \cite{symbolic} explored curvature adaptation in different contexts.

\section{Method}
\subsection{Core Algorithm}
SignCurv maintains three state variables per parameter:
\begin{itemize}
\item Momentum buffer $m_t$
\item Diagonal Hessian approximation $H_t$
\item Learning rate schedule $\eta_t$
\end{itemize}

\subsection{Algorithm Details}
The SignCurv optimizer proceeds as follows:
\begin{enumerate}
\item Initialize $m_0 = 0$, $H_0 = 0$
\item For each timestep $t$ from 1 to $T$:
  \begin{itemize}
  \item Compute gradients $g_t$
  \item Update momentum: $m_t = \beta_1 m_{t-1} + (1-\beta_1)g_t$
  \item Update Hessian: $H_t = \beta_2 H_{t-1} + (1-\beta_2)g_t^2$
  \item Compute sign update: $\Delta_t = -\eta_t \cdot \text{sign}(m_t) \odot (1 + \lambda H_t)^{-1}$
  \item Apply update: $\theta_{t+1} = \theta_t + \Delta_t$
  \end{itemize}
\end{enumerate}

\subsection{Hyperparameters}
Default values used in our experiments:
\begin{itemize}
\item $\beta_1 = 0.9$, $\beta_2 = 0.999$
\item $\lambda = 0.1$ (curvature scaling)
\item $\eta_{\text{max}} = 6e-4$, $\eta_{\text{min}} = 6e-5$
\item Warmup steps = 1000
\end{itemize}

\section{Experimental Setup}
We evaluate on the FineWeb dataset using:
\begin{itemize}
\item Model: Qwen architecture (134M parameters)
\item Batch size: 256
\item Sequence length: 2048
\item Training steps: 640
\item Hardware: 8x A100 GPUs
\end{itemize}

\section{Results}
\subsection{Main Results}
\begin{table}[h]
\centering
\caption{Comparison with baseline methods}
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & Relative Improvement \\
\midrule
AdamW & 4.927 & - \\
SignCurv (ours) & 4.018 & 18.4\% \\
Muon & 3.537 & - \\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[h]
  \centering
  \includegraphics[width=0.8\textwidth]{training_curves.png}
  \caption{Training dynamics showing stable optimization}
  \label{fig:training_curves}
\end{figure}

\subsection{Limitations}
\begin{itemize}
\item Does not surpass state-of-the-art Muon optimizer
\item Limited evaluation on single architecture/dataset
\item Higher memory usage than AdamW (41.8GB vs 31.5GB)
\end{itemize}

\section{Conclusion}
SignCurv demonstrates that combining sign-based updates with adaptive curvature information can yield competitive performance in transformer optimization. Future work should explore broader architectural support and hybrid approaches with methods like Muon.

\begin{thebibliography}{10}
\bibitem{adam} Kingma, Diederik P. and Ba, Jimmy. "Adam: A Method for Stochastic Optimization." \textit{arXiv preprint arXiv:1412.6980}, 2014.

\bibitem{adamw} Loshchilov, Ilya and Hutter, Frank. "Decoupled Weight Decay Regularization." \textit{arXiv preprint arXiv:1711.05101}, 2017.

\bibitem{signsgd} Bernstein, Jeremy et al. "signSGD: Compressed Optimisation for Non-Convex Problems." \textit{arXiv preprint arXiv:1802.04434}, 2018.

\bibitem{shampoo} Anil, Rohan et al. "Scalable Second Order Optimization for Deep Learning." \textit{arXiv preprint arXiv:2002.09018}, 2020.

\bibitem{muon} "Muon Optimizer." AardXiv 2510.00111, 2025.

\bibitem{velo} "VeLO: Training Versatile Learned Optimizers." AardXiv 2511.00024, 2025.

\bibitem{symbolic} "Symbolic Discovery of Optimizers." AardXiv 2511.00013, 2025.

\bibitem{practical} "Practical Tradeoffs in Optimizer Design." AardXiv 2510.00052, 2025.
\end{thebibliography}

\end{document}
