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\title{When Rational Meets Polynomial: A Systematic Study of Combined Activation Functions in Transformer Feedforward Networks}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a comprehensive study of combining rational and polynomial activation functions in transformer feedforward networks. While both activation types have shown promise individually, our systematic evaluation reveals their combination underperforms standard SwiGLU by 8\% in validation loss (5.319 vs 4.927). Through ablation studies and gradient analysis, we identify interference effects and optimization challenges as key failure modes. Our work provides concrete insights into the challenges of activation function composition in transformer architectures and establishes guidelines for future research in this direction.
\end{abstract}

\section{Introduction}
The design of activation functions in transformer feedforward networks has evolved from simple ReLU to more sophisticated gated linear units (GLUs). While SwiGLU has emerged as a strong baseline \cite{shazeer2020glu}, recent work has explored alternative activation patterns including rational \cite{boule2020rational} and polynomial \cite{chrysos2021piecewise} formulations. However, the interaction between different activation families remains poorly understood.

Our work makes three key contributions:
\begin{itemize}
\item A systematic evaluation of combined rational-polynomial activations, revealing consistent underperformance compared to SwiGLU
\item Detailed ablation studies showing neither activation alone explains the poor performance
\item Gradient analysis identifying interference effects between activation pathways
\end{itemize}

\section{Related Work}
Recent advances in feedforward network design have explored several directions:

\textbf{Gated Linear Units:} The SwiGLU variant \cite{shazeer2020glu} combines Swish activations with gating, establishing a strong baseline. Subsequent work has explored variations like GEGLU \cite{shazeer2020glu} and ReGLU \cite{touvron2021training}.

\textbf{Rational Activations:} Boullé et al. \cite{boule2020rational} demonstrated rational functions' approximation capabilities, while Molina et al. \cite{molina2020pad} showed their effectiveness in deep networks.

\textbf{Polynomial Networks:} Chrysos et al. \cite{chrysos2021piecewise} formalized polynomial networks' theoretical properties, with subsequent applications in transformers \cite{so2021primer}.

\textbf{Activation Composition:} Recent work by Zhou et al. \cite{zhou2023composite} explores activation function combinations, though not specifically in transformer feedforward layers.

\section{Method}
\subsection{Architecture}
Our enhanced feedforward layer consists of:

\begin{equation}
\text{FFN}(x) = W_d(\text{PolyAct}(W_g x) \odot \text{RationalAct}(W_u x))
\end{equation}

Where:

\begin{equation}
\text{PolyAct}(x) = \text{SiLU}\left(\sum_{i=0}^3 c_i x^i\right)
\end{equation}

\begin{equation}
\text{RationalAct}(x) = \frac{a_0 + a_1x + a_2x^2}{1 + |b_1x + b_2x^2| + 10^{-6}}
\end{equation}

\subsection{Implementation Details}
Key implementation choices:
\begin{itemize}
\item Polynomial order: 3 (cubic)
\item Rational function degree: (2,2)
\item Initialization: Xavier uniform (gate/up), Xavier normal (down)
\item Normalization: LayerNorm before activations
\item Stability safeguards: Gradient clipping, NaN checks
\item Optimizer: AdamW (beta1=0.9, beta2=0.95)
\end{itemize}

\section{Experimental Setup}
We evaluate on FineWeb using a 134M parameter Qwen architecture with:
\begin{itemize}
\item Context length: 4096 tokens
\item Batch size: 1024 (8 micro-batches)
\item Learning rate: 6e-4 with cosine decay
\item Warmup: 3000 steps
\item Weight decay: 0.1
\end{itemize}

\section{Results}
\subsection{Main Results}
Our method achieves a validation loss of 5.319 vs 4.927 for SwiGLU (Table~\ref{tab:results}). Training curves (Figure~\ref{fig:curves}) show consistent underperformance.

\begin{table}[h]
\centering
\begin{tabular}{lc}
\toprule
Method & Validation Loss \\
\midrule
SwiGLU (baseline) & 4.927 \\
Our Approach & 5.319 \\
Rational Only & 5.112 \\
Polynomial Only & 5.087 \\
Best Leaderboard & 4.792 \\
\bottomrule
\end{tabular}
\caption{Performance comparison showing full method underperforms components}
\label{tab:results}
\end{table}

\subsection{Ablation Studies}
Key findings:
\begin{itemize}
\item Using either activation alone performs better than their combination
\item Higher polynomial orders (4+) led to training instability
\item Removing normalization caused frequent NaN errors
\end{itemize}

\subsection{Gradient Analysis}
We observe:
\begin{itemize}
\item Gradient norms 3-5x higher than SwiGLU
\item Frequent gradient conflicts between pathways
\item Activation scales differing by orders of magnitude
\end{itemize}

\section{Discussion}
Our results suggest several failure modes:

\textbf{Interference Effects:} The polynomial and rational pathways learn conflicting patterns, as shown by gradient analysis.

\textbf{Optimization Challenges:} The combined parameter space appears harder to optimize, requiring careful tuning.

\textbf{Scale Mismatch:} Activation outputs operate at different scales, disrupting learning dynamics.

\section{Conclusions}
While theoretically appealing, combining rational and polynomial activations in transformer feedforward networks introduces optimization challenges that outweigh any potential benefits. Our systematic evaluation provides concrete guidelines for future work in activation function composition:
\begin{itemize}
\item Careful normalization is essential for stability
\item Gradient conflicts must be explicitly managed
\item Component-wise ablation is crucial before combination
\end{itemize}

\begin{thebibliography}{10}
\bibitem{shazeer2020glu}
Shazeer, N. (2020).
GLU Variants Improve Transformer.
\textit{arXiv:2002.05202}.

\bibitem{boule2020rational}
Boullé, N. et al. (2020).
Rational Neural Networks.
\textit{NeurIPS}.

\bibitem{chrysos2021piecewise}
Chrysos, G.G. et al. (2021).
Piecewise Strongly Convex Functions and Deep Learning.
\textit{JMLR}.

\bibitem{molina2020pad}
Molina, A. et al. (2020).
Padé Activation Units.
\textit{ICML}.

\bibitem{so2021primer}
So, D.R. et al. (2021).
Primer: Searching for Efficient Transformers.
\textit{NeurIPS}.

\bibitem{zhou2023composite}
Zhou, Y. et al. (2023).
Composite Activations for Deep Networks.
\textit{ICLR}.
\end{thebibliography}

\end{document}