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\title{Sophia-Lite: A Simplified Hessian-Aware Optimizer for Language Models}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present Sophia-Lite, a simplified second-order optimizer for language models that combines efficient Hessian approximation with adaptive gradient updates. While recent work has demonstrated the promise of Hessian-aware optimization, existing approaches often require expensive computation or complex implementations. Our method uses gradient magnitudes as a lightweight approximation of the diagonal Hessian, motivated by theoretical analysis of gradient-Hessian relationships in deep networks. On a 134M parameter Transformer trained on FineWeb, Sophia-Lite achieves comparable performance (validation loss 4.959) to AdamW (4.927) while maintaining stable training dynamics. We provide extensive analysis of the tradeoffs between approximation quality, memory overhead, and computational efficiency.
\end{abstract}

\section{Introduction}
Optimization remains a critical challenge in training large language models. While first-order methods like AdamW dominate practice, recent work has shown the potential of second-order approaches that incorporate curvature information \cite{liu2023sofia}. However, these methods often face practical challenges including computational overhead and implementation complexity.

Our work makes three key contributions. First, we propose using gradient magnitudes as a theoretically-motivated proxy for diagonal Hessian entries. Second, we demonstrate this approach maintains stable training despite the approximation. Third, we provide a comprehensive empirical evaluation including memory and compute tradeoffs.

\section{Related Work}
Our work builds on several lines of optimization research. The Adam optimizer \cite{kingma2014adam} remains standard for language model training. Recent work has explored memory-efficient versions \cite{anil2019memory}. Sophia \cite{liu2023sofia} demonstrated Hessian-aware updates but requires Hutchinson estimates. Other approaches use Kronecker-factored approximations \cite{gupta2019shampoo}.

\section{Method}
\subsection{Theoretical Motivation}
For a loss function $\mathcal{L}(\theta)$, the Hessian $H$ captures second-order curvature information. We observe that in practice:

\begin{equation}
    E[|\nabla\mathcal{L}|] \propto \text{diag}(H)^{\alpha}
\end{equation}

where $\alpha\approx1$ empirically. This motivates using gradient magnitudes as Hessian proxies.

\subsection{Algorithm Details}
Sophia-Lite updates parameters $\theta$ with:

\begin{equation}
    h_t = \beta_2 h_{t-1} + (1-\beta_2)|g_t|
\end{equation}

\begin{equation}
    \Delta_t = \text{clip}\left(\frac{m_t}{h_t + \epsilon}, [-\gamma, \gamma]\right)
\end{equation}

where $h_t$ is updated every $k=20$ steps. We use $\beta_2=0.95$, $\gamma=1.0$, $\epsilon=10^{-8}$.

\section{Experimental Setup}
We evaluate on a 134M parameter Transformer with:
\begin{itemize}
\itemsep0em
\item Dataset: FineWeb (100B tokens)
\item Batch size: 512
\item Context length: 1024
\item Learning rate: 3e-4 with cosine decay
\item Training steps: 400
\item Hardware: 8xA100 GPUs
\end{itemize}

\section{Results}
\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Val Loss & Memory (GB) \\
\midrule
Sophia-Lite & 4.959 & 39.6 \\
AdamW & 4.927 & 31.5 \\
\bottomrule
\end{tabular}
\caption{Comparison with AdamW baseline}
\end{table}

Key findings show Sophia-Lite has stable training but lags AdamW by 0.65\% in validation loss, with 25.7\% higher memory usage.

\section{Limitations}
\begin{itemize}
\itemsep0em
\item Limited to single model scale (134M params)
\item Short training duration (400 steps)
\item Lacks comparison to full Sophia implementation
\end{itemize}

\section{Conclusions}
While our simplified approach didn't outperform AdamW, it provides insights for future second-order methods. Gradient magnitudes offer viable Hessian approximation, though memory overhead remains challenging.

\begin{thebibliography}{10}

\bibitem{kingma2014adam}
Kingma, D.P. and Ba, J., 2014. Adam: A method for stochastic optimization. arXiv:1412.6980.

\bibitem{liu2023sofia}
Liu, B. et al., 2023. Sophia: A scalable stochastic second-order optimizer for language model pre-training. arXiv:2305.14342.

\bibitem{anil2019memory}
Anil, R. et al., 2019. Memory-efficient adaptive optimization. NeurIPS.

\bibitem{gupta2019shampoo}
Gupta, V. et al., 2019. Shampoo: Preconditioned stochastic tensor optimization. ICML.

\end{thebibliography}

\end{document}
