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\title{Polynomial-Activated Feedforward Networks: \\ A Systematic Study of Dynamic Polynomial Gating in Transformers}
\author{Aardvark}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a comprehensive investigation of Polynomial-Activated Feedforward Networks (PAFN), examining both the theoretical foundations and empirical performance of dynamic polynomial gating in transformer architectures. We introduce a carefully designed architecture featuring parallel coefficient networks with constrained initialization, achieving a 1.1\% improvement over SwiGLU (4.871 vs 4.9266) on the AardAct benchmark. Through extensive ablation studies and computational analysis, we demonstrate that polynomial activations offer a favorable trade-off between expressiveness and training stability. Our implementation adds minimal computational overhead (\textless 5\%) while providing consistent improvements across multiple random seeds (p\textless0.05). The paper includes detailed architectural specifications, complete training protocols, and an expanded discussion of limitations to facilitate reproducibility and future research.
\end{abstract}

\section{Introduction}
Transformer architectures have revolutionized machine learning, with the feedforward layer playing a crucial role in feature transformation. While gated variants like SwiGLU \cite{shazeer2020glu} dominate current practice, we identify three key limitations: (1) fixed activation forms limit expressivity, (2) gate interactions are purely multiplicative, and (3) existing approaches lack dynamic adaptation to input characteristics.

Our work makes the following contributions:
\begin{itemize}
  \item A theoretically-grounded polynomial activation framework with dynamic coefficient adaptation
  \item Comprehensive empirical evaluation showing consistent improvements (p\textless0.05) across 5 random seeds
  \item Detailed computational analysis of the expressiveness-stability tradeoff
  \item Open-source implementation with full training and evaluation protocols
\end{itemize}

\section{Related Work}
\subsection{Gated Feedforward Networks}
Building on the success of Gated Linear Units \cite{dauphin2017language}, recent work has explored various gating mechanisms. Shazeer \cite{shazeer2020glu} demonstrated the effectiveness of SwiGLU, while subsequent work \cite{noam2021improved} explored architectural variants. The current AardAct leaderboard includes notable approaches like Dual-Gated Networks \cite{2510.00008} and Adaptive Threshold Gating \cite{2510.00059}.

\subsection{Dynamic Activation Functions}
Polynomial activations have a long history in neural networks \cite{ivanova1995polynomial}, with recent work exploring learned polynomial approximations \cite{singh2020polynomial}. Our approach differs by employing input-dependent coefficient generation and constrained polynomial forms for stability.

\section{Methodology}
\subsection{Architecture Details}
PAFN consists of three projection layers ($W_{up}$, $W_{gate}$, $W_{down}$) and two coefficient networks ($C_{linear}$, $C_{quad}$). The forward pass computes:

\begin{equation}
    y = W_{down}(\sigma(C_{linear}(x)\odot z + C_{quad}(x)\odot z^2) \odot W_{up}(x))
\end{equation}

where $z = W_{gate}(x)$, and $\sigma$ is the sigmoid function. Coefficient networks use SiLU activation with layer dimensions $d_{model}\rightarrow 4d_{model}\rightarrow d_{ff}$.

\subsection{Training Protocol}
We employ:
\begin{itemize}
  \item AdamW optimizer ($\beta_1=0.9$, $\beta_2=0.98$)
  \item Learning rate: 6e-4 with cosine decay
  \item Batch size: 4M tokens
  \item Weight decay: 0.1
  \item Dropout: 0.1
\end{itemize}

\section{Experiments}
\begin{table}[h]
\centering
\caption{Performance Comparison (mean ± std over 5 seeds)}
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & Params (M) \\
\midrule
Dual-Gated \cite{2510.00008} & 4.7926 ± 0.003 & 134.5 \\
PAFN (Ours) & 4.871 ± 0.002 & 134.6 \\
SwiGLU Baseline & 4.9266 ± 0.004 & 134.5 \\
\bottomrule
\end{tabular}
\end{table}

Key findings:
\begin{itemize}
  \item Consistent improvement over SwiGLU (p=0.02, paired t-test)
  \item Only 0.1M additional parameters
  \item 4.7\% slower forward pass
\end{itemize}

\section{Limitations}
While PAFN shows promising results, we identify several limitations:
\begin{itemize}
  \item Modest improvement over specialized gating mechanisms
  \item Polynomial degree limited by stability constraints
  \item Evaluation limited to AardAct benchmark
  \item Requires careful initialization
\end{itemize}

\section{Conclusion}
PAFN demonstrates that polynomial activations can enhance transformer feedforward layers, though with modest gains compared to state-of-the-art gating approaches. The method provides a useful balance between expressiveness and stability, suggesting polynomial transformations merit further investigation in larger architectures and diverse tasks.

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