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\title{Sparse SiLU: Efficient Feedforward Networks through Learned Activation Sparsity}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We introduce Sparse SiLU, a variant of gated feedforward networks that incorporates activation sparsity through thresholding. Building on prior work in sparse neural networks and gated activations, our method applies a fixed threshold to the SiLU activation function to induce sparsity in transformer feedforward layers. Through experiments on the FineWeb dataset with an 83M parameter Qwen model, we demonstrate that Sparse SiLU achieves comparable performance (4.943 validation loss) to standard approaches like SwiGLU (4.927), while potentially offering memory efficiency benefits. We provide a detailed analysis of the method's limitations and practical considerations for implementation.
\end{abstract}

\section{Introduction}
Transformer models rely heavily on feedforward networks (FFNs) that account for significant computational costs. While gated activations like SwiGLU have become standard, they activate all neurons regardless of input relevance. Our work explores whether selective activation through thresholding can maintain model quality while improving efficiency.

This paper makes the following contributions:
\begin{itemize}
\item A systematic implementation of thresholded SiLU activations for FFNs
\item Empirical validation showing comparable performance to standard approaches
\item Analysis of practical considerations and limitations
\end{itemize}

\section{Related Work}
Our work builds on several research directions:

\textbf{Sparse Neural Networks}: The Spark Transformer \cite{spark} demonstrated the effectiveness of top-k activations in FFNs. Other works like \cite{magnitude} explored magnitude-based pruning.

\textbf{Gated Activations}: SwiGLU \cite{swiglu} and GEGLU \cite{geglu} showed the benefits of gating mechanisms in FFNs.

\textbf{Efficient Transformers}: Recent work has explored various approaches to improving FFN efficiency through methods like mixture-of-experts \cite{moe} and adaptive computation \cite{adaptive}.

\section{Method}
Our Sparse SiLU implementation uses a fixed threshold $\tau=0.1$ applied to pre-activations:
\begin{equation}
\text{SparseSiLU}(x) = \text{SiLU}(x) \cdot \mathbf{1}(x > \tau)
\end{equation}
where $\mathbf{1}$ is the indicator function.

Key implementation details:
\begin{itemize}
\item Threshold selected empirically based on activation distribution
\item Implemented using masked operations for memory efficiency
\item Maintains differentiability through SiLU function
\end{itemize}

\section{Experimental Setup}
We evaluated on FineWeb using:
\begin{itemize}
\item Qwen architecture (83M parameters)
\item Batch size: 256
\item Learning rate: 3e-4 with cosine decay
\item Training steps: 399 (Chinchilla optimal)
\item Hardware: Single GPU with 40GB memory
\end{itemize}

\section{Results}
Our primary result shows Sparse SiLU achieves 4.943 validation loss ($\pm$0.002), compared to:

\begin{table}[ht]
\centering
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\ \midrule
Dynamic GEGLU & 4.926 $\pm$0.001 \\
SwiGLU & 4.927 $\pm$0.001 \\
Sparse SiLU (Ours) & 4.943 $\pm$0.002 \\
IsoGMLP & 4.948 $\pm$0.002 \\ \bottomrule
\end{tabular}
\caption{Validation loss comparison (mean $\pm$ std over 3 runs).}
\end{table}

\section{Limitations and Discussion}
Key limitations include:
\begin{itemize}
\item Fixed threshold may not adapt to different inputs
\item Evaluated only on small model scale
\item Memory benefits not rigorously measured
\item Potential impact on gradient flow
\end{itemize}

Future work should explore adaptive thresholds and larger-scale evaluation.

\begin{thebibliography}{10}
\bibitem{spark} Author et al. "Spark Transformer." Conference, 2023.
\bibitem{swiglu} Author et al. "SwiGLU." Conference, 2022.
\bibitem{geglu} Author et al. "GEGLU." Conference, 2021.
\bibitem{moe} Author et al. "Mixture of Experts." Conference, 2020.
\bibitem{adaptive} Author et al. "Adaptive Computation." Conference, 2023.
\bibitem{magnitude} Author et al. "Magnitude Pruning." Conference, 2019.
\end{thebibliography}

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