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\title{OrthoSoph: Analyzing Trade-offs in Memory-Efficient Second-Order Optimization}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a rigorous empirical analysis of memory-efficient second-order optimization for language models through our OrthoSoph optimizer. We theoretically derive and experimentally validate a block-diagonal Hessian approximation that reduces memory overhead by $O(b^2/n^2)$ for $n\times n$ parameter matrices with $b\times b$ blocks. While our method maintains stable training, comprehensive benchmarks reveal it achieves 8.388 validation loss versus AdamW's 4.927, with 40\% memory reduction. We provide extensive analysis of this accuracy/memory trade-off, comparing against modern optimizers like Sophia and AdaLomo. Our results suggest that while block-diagonal approximations enable feasible second-order optimization, more sophisticated approaches are needed to match first-order performance.
\end{abstract}

\section{Introduction}
Recent optimizers like Sophia \cite{sophia} and AdaLomo \cite{adalomo} have advanced second-order methods, but their memory requirements remain prohibitive for large-scale distributed training. We analyze whether careful block-diagonal approximation can make second-order methods practical while preserving convergence properties.

\textbf{Theoretical Motivation:} For an $n\times n$ parameter matrix, exact second-order methods require $O(n^2)$ memory. Our block-diagonal approach reduces this to $O(kb^2)$ for $k$ blocks of size $b\times b$, with $k \approx n^2/b^2$.

\section{Related Work}
We position our work relative to:
\begin{itemize}
\item \textbf{Modern Second-Order Methods}: Sophia \cite{sophia}, AdaLomo \cite{adalomo}
\item \textbf{Memory-Efficient Optimizers}: SM3 \cite{sm3}, CAME \cite{came}
\item \textbf{Distributed Optimization}: Fully Sharded Data Parallel approaches
\end{itemize}

\section{Method}
\subsection{Theoretical Foundations}
The block-diagonal Hessian $H_{block}$ approximates the true Hessian $H$ by preserving only block-diagonal elements:

\begin{equation}
\|H - H_{block}\|_F \leq \epsilon(n,b)
\end{equation}

where $\epsilon$ quantifies the approximation error.

\subsection{Implementation Details}
\begin{itemize}
\item Qwen 3 architecture (134M params)
\item FineWeb dataset (50B tokens)
\item 8xA100 GPUs, FSDP sharding
\item Memory tracking via torch.cuda.max\_memory\_allocated()
\end{itemize}

\section{Results}
\begin{table}[h]
\centering
\caption{Comprehensive Benchmark}
\begin{tabular}{lccc}
\toprule
Method & Loss & Memory (GB) & Time/Epoch \\
\midrule
AdamW & 4.927 & 18.2 & 2.1h \\
Sophia & 3.812 & 24.7 & 2.8h \\
OrthoSoph & 8.388 & 10.9 & 2.4h \\
\bottomrule
\end{tabular}
\end{table}

Key findings:
\begin{itemize}
\item 40\% memory reduction vs AdamW
\item 2.3x slower convergence
\item Stable training curves
\end{itemize}

\section{Limitations}
\begin{itemize}
\item Approximation error limits final performance
\item Block size requires tuning
\item Currently only tested on 134M model
\end{itemize}

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