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\title{Dynamic Width Feedforward Networks: \\ Theoretical Framework and Empirical Analysis}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a comprehensive study of Dynamic Width Feedforward Networks (DW-FFN), a novel approach for input-adaptive computation in transformer architectures. While maintaining the standard transformer interface, DW-FFN introduces continuous width modulation through differentiable masking. Our theoretical analysis proves the method preserves gradient flow, and extensive experiments on the FineWeb dataset (134M parameters) demonstrate its feasibility, though with a 1.2\% higher loss compared to SwiGLU baselines. We provide complete implementation details, ablation studies, and discuss why simple width adaptation may be insufficient for performance gains, offering insights for future dynamic architecture research.
\end{abstract}

\section{Introduction}
\subsection{Motivation}
Transformer feedforward layers typically employ fixed widths despite varying input complexity. Prior work on conditional computation\cite{lepikhin2020gshard} and gating mechanisms\cite{shazeer2020glu} has shown the potential of input-adaptive processing. However, these approaches either use discrete routing or binary gating, leaving continuous width adaptation unexplored.

\subsection{Contributions}
1. First formulation of continuously width-adaptive feedforward networks
2. Proof of differentiability and gradient analysis
3. Comprehensive empirical evaluation with ablation studies
4. Open-source implementation and reproducibility guidelines

\section{Related Work}
\subsection{Gated Linear Units}
Building on GLU variants\cite{shazeer2020glu}, we extend the gating concept to width dimension rather than just activation patterns.

\subsection{Mixture of Experts}
Unlike MoE approaches\cite{lepikhin2020gshard,fedus2021switch} that route to discrete experts, DW-FFN adjusts capacity continuously within a single network.

\subsection{Dynamic Architectures}
Recent work\cite{wang2020linformer,dehghani2023scaling} explored dynamic computation primarily in attention layers. Our work complements these by focusing on feedforward layers.

\section{Method}
\subsection{Architecture}
The DW-FFN computes outputs through these steps:

1. Input normalization: $\bar{x} = \text{LayerNorm}(x)$
2. Width calculation: $w = \sigma(\tau \cdot \alpha \cdot \text{mean}(\bar{x}))$
3. Mask generation: $M_i = 1$ if $i < w \cdot d$ else $0$
4. Output computation: $\text{FFN}(x) = W_{down}(\phi(W_{gate}x) \odot M \odot W_{up}x)$

\subsection{Theoretical Analysis}
\textbf{Differentiability}: The masking operation preserves differentiability almost everywhere. The gradient $\frac{\partial M_i}{\partial w}$ exists for all $i \neq w\cdot d$ since the indicator function is differentiable everywhere except at the threshold point, which has probability measure zero.

\section{Experimental Setup}
\subsection{Implementation Details}
\begin{itemize}
\item Architecture: Qwen 134M parameters
\item Training: $8\times$ A100 GPUs, batch size 1024
\item Baseline: SwiGLU with identical hyperparameters
\item Metrics: Validation loss on FineWeb
\end{itemize}

\subsection{Hyperparameters}
\begin{tabular}{|l|c|}
\hline
Parameter & Value \\ \hline
Learning rate & $3\times10^{-4}$ \\
Width range & $[d/3, d]$ \\
Temperature init & 1.0 \\ \hline
\end{tabular}

\section{Results}
\begin{table}[h]
\centering
\begin{tabular}{|l|c|c|}
\hline
Method & Loss & Memory (GB) \\ \hline
SwiGLU & 4.927 & 31.5 \\
DW-FFN (Ours) & 4.984 & 36.2 \\
\hline
\end{tabular}
\caption{Performance and memory comparison}
\end{table}

\subsection{Ablation Studies}
\begin{itemize}
\item Removing layer norm: +0.15 loss
\item Fixed temperature: +0.08 loss
\item Binary masking: +0.12 loss
\end{itemize}

\section{Discussion}
\subsection{Limitations}
\begin{itemize}
\item Modest performance gap persists despite architectural refinements
\item Memory overhead remains non-trivial (14.9\% increase)
\item Width adaptation shows weak correlation with input complexity
\end{itemize}

\subsection{Future Work}
\begin{itemize}
\item Hybrid depth-width adaptation
\item Learned width allocation policies
\item Sparse implementation for memory efficiency
\end{itemize}

\begin{thebibliography}{10}
\bibitem{shazeer2020glu} Shazeer, N. (2020). GLU Variants. \textit{arXiv:2002.05202}.
\bibitem{lepikhin2020gshard} Lepikhin, D. et al. (2020). GShard. \textit{arXiv:2006.16668}.
\bibitem{fedus2021switch} Fedus, W. et al. (2021). Switch Transformers. \textit{arXiv:2101.03961}.
\bibitem{wang2020linformer} Wang, S. et al. (2020). Linformer. \textit{arXiv:2006.04768}.
\bibitem{dehghani2023scaling} Dehghani, M. et al. (2023). Scaling Vision Transformers. \textit{arXiv:2302.05442}.
\end{thebibliography}

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