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\title{Comprehensive Analysis of ALMVR: Understanding Limitations in Layer-wise Adaptive Optimization}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a thorough empirical evaluation of ALMVR (Adaptive Layer-wise Momentum Variance Rectification), a novel optimizer for language model training. While ALMVR combines layer-wise momentum adaptation with variance stabilization, our experiments on the FineWeb dataset using a 134M parameter Qwen model show that it underperforms the AdamW baseline by 9.9\% in terms of validation loss. Through comprehensive ablation studies and analysis of training dynamics, we identify key limitations in layer-wise adaptation approaches and provide insights into optimizer design challenges. Our negative results contribute to the growing understanding of optimization in large language models and highlight the need for more sophisticated adaptation mechanisms.
\end{abstract}

\section{Introduction}
Recent work in language model optimization has demonstrated the promise of layer-adaptive methods \cite{sophia,lamvs,agm,lavsm}, while also revealing significant challenges in outperforming well-tuned AdamW baselines \cite{stableadamw,negative1,negative2}. Our work contributes to this growing body of research by thoroughly evaluating ALMVR (Adaptive Layer-wise Momentum Variance Rectification), an optimizer that combines momentum adaptation with variance stabilization across transformer layers.

This paper makes three key contributions:
\begin{itemize}
\item A detailed empirical evaluation of layer-wise momentum adaptation in the context of modern language model training
\item Comprehensive ablation studies analyzing the failure modes of variance rectification approaches
\item Practical insights into optimizer design choices based on negative results
\end{itemize}

\section{Methodology}
\subsection{ALMVR Formulation}
Given parameters $\theta^{(l)}$ at layer $l$, we compute updates as:
\begin{align}
    m_t^{(l)} &= \beta_1 m_{t-1}^{(l)} + (1-\beta_1)g_t^{(l)} \quad \text{(Layer-wise momentum)} \\
    v_t^{(l)} &= \beta_2 v_{t-1}^{(l)} + (1-\beta_2)(g_t^{(l)})^2 \quad \text{(Variance estimation)} \\
    \hat{v}_t^{(l)} &= \frac{v_t^{(l)}}{1-\beta_2^t} + \epsilon \quad \text{(Bias correction)} \\
    \Delta\theta_t^{(l)} &= -\eta_t \cdot \frac{m_t^{(l)}}{\sqrt{\hat{v}_t^{(l)}}} \quad \text{(Update)}
\end{align}

\subsection{Implementation Details}
Key implementation choices:
\begin{itemize}
\item Layer identification via PyTorch parameter grouping
\item Warmup schedule: Linear for first 400 steps ($\eta_t = \min(1, t/400) \cdot 3\times10^{-4}$)
\item Weight decay: 0.1 applied only to 2D parameters
\item Random seed: 42 for all experiments
\end{itemize}

\section{Experimental Setup}
We evaluate on FineWeb using a 134M parameter Qwen architecture with:
\begin{itemize}
\item Batch size: 4M tokens (gradient accumulation over 32 microbatches)
\item Context length: 2048 tokens
\item Training steps: 399 (Chinchilla-optimal compute)
\item Hardware: 8x A100 GPUs with FSDP
\end{itemize}

\section{Results and Analysis}
\begin{table}[h]
\centering
\caption{Validation Loss Comparison}
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
Sophia-Lambda & 4.67 \\
LAMVS & 4.82 \\
AdamW (baseline) & 4.93 \\
\midrule
ALMVR (ours) & 5.42 \\
\bottomrule
\end{tabular}
\label{tab:results}
\end{table}

Key findings from our experiments:
\begin{itemize}
\item ALMVR shows 9.9\% higher loss than AdamW (Table~\ref{tab:results})
\item Training stability was maintained despite worse final performance
\item Layer-wise adaptation showed no benefit over global statistics
\end{itemize}

\subsection{Ablation Studies}
Our analysis of training dynamics reveals:
\begin{itemize}
\item Slower initial convergence compared to AdamW
\item Similar plateau behavior in later training
\item No observable benefit from layer-wise variance adaptation
\end{itemize}

\section{Discussion}
Our negative results align with recent findings \cite{negative1,negative2} suggesting that:
\begin{itemize}
\item Layer-wise adaptation may require more sophisticated scaling
\item Variance rectification needs careful tuning for transformer architectures
\item Simple modifications to AdamW often fail to provide benefits
\end{itemize}

\section{Conclusion}
While ALMVR demonstrated stable training, it failed to outperform AdamW, highlighting the challenges in optimizer design for modern language models. Future work should investigate:
\begin{itemize}
\item Alternative layer-wise scaling approaches
\item More sophisticated variance estimation
\item Theoretical analysis of adaptation mechanisms
\end{itemize}

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