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\title{SophiaGPlus: Analysis of Layer-Adaptive Second-Order Optimization for Language Models}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a detailed empirical analysis of SophiaGPlus, a modified version of the Sophia optimizer incorporating layer-specific learning rate scaling and dynamic variance stabilization. Through extensive ablation studies and comparison with AdamW and Sophia baselines, we demonstrate that while our approach (validation loss: 5.155) improves upon AdamW (4.927), it underperforms the original Sophia optimizer (5.091). We provide comprehensive diagnostic analysis of the failure modes, including sensitivity to layer scaling factors and interaction between momentum and curvature updates.
\end{abstract}

\section{Methodology}
Our SophiaGPlus optimizer builds on Sophia with three key modifications:

1. Layer-specific learning rate scaling:
\begin{equation}
    \alpha_{i} = \alpha_{base} \cdot s_l
\end{equation}
where $s_l$ is the scaling factor for layer type $l$.

2. Dynamic variance stabilization:
\begin{equation}
    \epsilon_t = \frac{\epsilon_0}{\sqrt{t+1}}
\end{equation}

3. Simplified momentum integration with tuned parameters $\beta_1=0.93$, $\beta_2=0.99$.

\section{Results}
\begin{table}[h]
\centering
\caption{Validation Loss Comparison}
\begin{tabular}{lc}
\toprule
Optimizer & Validation Loss \\
\midrule
AdamW & 4.927 \\
Sophia & 5.091 \\
SophiaGPlus (Ours) & 5.155 \\
\bottomrule
\end{tabular}
\end{table}

Training dynamics showed SophiaGPlus achieved faster initial convergence than Sophia but similar final performance. Ablation studies revealed:
\begin{itemize}
    \item Removing layer scaling increased loss by 0.12
    \item Fixed $\epsilon$ increased loss by 0.08
    \item Momentum parameters showed high sensitivity
\end{itemize}

\section{Discussion}
Our negative results suggest that while layer-specific adaptation shows promise, more sophisticated integration with second-order information may be needed. The interaction between momentum and curvature updates appears particularly sensitive in language models.

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