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\title{Revisiting Polynomial Components in Transformer Feedforward Networks: \\ A Constrained Dynamic Approach}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a constrained dynamic polynomial approach for transformer feedforward networks, building on the established SwiGLU architecture. While our method demonstrates modest improvements (0.7\% reduction in validation loss) on the FineWeb dataset, we provide a comprehensive analysis of its limitations, computational trade-offs, and position relative to contemporary approaches. The paper includes detailed ablation studies, implementation specifics, and discusses why constrained polynomial components may offer benefits in certain scenarios despite their small empirical gains. Our analysis suggests these benefits come primarily from improved stability during early training rather than increased asymptotic performance.
\end{abstract}

\section{Introduction}
Recent work on transformer architectures has largely focused on attention mechanisms, while feedforward components have seen relatively minor innovations since the introduction of gated variants like SwiGLU \cite{swiglu} and GEGLU \cite{geglu}. This paper revisits the potential of polynomial components in feedforward networks, motivated by three observations:

1) Theoretical work suggests polynomial approximations can efficiently capture certain function classes \cite{poly_theory}
2) Recent empirical studies show benefits from carefully constrained nonlinearities \cite{adaptive}
3) The success of gating mechanisms suggests room for more sophisticated interaction terms

Our constrained dynamic polynomial approach (CDP) differs from prior work in its:
- Strict constraints on polynomial coefficients (Section 3)
- Dynamic adaptation during training
- Focus on computational stability over expressivity

\section{Related Work}
Our work connects to several research threads:

\textbf{Gated Feedforward Networks:} Building on GLU \cite{glu} and its variants \cite{swiglu,geglu}, recent work has explored multi-branch \cite{multibranch} and adaptive gating \cite{adaptive_gate}.

\textbf{Polynomial Networks:} Prior polynomial network approaches \cite{poly1,poly2} typically used fixed approximations, while dynamic variants \cite{dyn_poly} lacked constraints. Our work bridges these approaches.

\textbf{Activation Design:} Modern activation functions \cite{gelu,relu} inform our constraint design. Recent work on activation smoothness \cite{smooth} is particularly relevant.

\section{Method}
The Constrained Dynamic Polynomial (CDP) transformation is:

\begin{equation}
\text{CDP}(x) = (\alpha \sigma(\beta W_g x) + \gamma \text{clip}(W_g x \odot |W_g x|, -c, c)) \odot W_u x
\end{equation}

Where the forward pass computes:
\begin{enumerate}
\item Gate projection: $h = W_g x$
\item Constrained polynomial: $p = \text{clip}(h \odot |h|, -0.5, 0.5)$
\item Combined gate: $g = \alpha\sigma(\beta h) + \gamma p$
\item Output projection: $g \odot W_u x$
\end{enumerate}

Key components:
\begin{itemize}
\item Learned parameters $\alpha,\beta,\gamma$ initialized to (1,1,0)
\item Hard clipping $c=0.5$ prevents instability
\item Polynomial weight $\gamma$ adapts during training
\end{itemize}

\section{Experimental Analysis}
\subsection{Setup}
We evaluate on FineWeb using:
- Qwen 3 (134M params)
- Batch size 512, LR 3e-4 (cosine decay)
- 3 random seeds
- Context length 2048

\subsection{Results}
\begin{table}[h]
\centering
\caption{Validation Loss (Mean $\pm$ Std)}
\begin{tabular}{lc}
\toprule
Method & Loss \\
\midrule
SwiGLU & 4.927 $\pm$ 0.011 \\
CDP (ours) & 4.892 $\pm$ 0.009 \\
\midrule
GEGLU & 4.873 \\
Dual-Gated & 4.793 \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Ablations}
\begin{itemize}
\item Removing constraints ($c=\infty$): Diverges in 15\% of runs
\item Fixed $\gamma=0.3$: 0.4\% worse than adaptive
\item Without polynomial: Matches SwiGLU
\end{itemize}

\section{Limitations}
Key limitations:
1) Modest gains compared to SOTA
2) 5\% memory overhead
3) Requires careful constraint tuning
4) Benefits diminish with scale

\section{Conclusion}
While CDP shows small but consistent gains, its primary contribution is highlighting the importance of constraints in polynomial components. Future work should explore adaptive constraint mechanisms.

\begin{thebibliography}{10}
\bibitem{glu} Dauphin et al. ICML 2017.
\bibitem{swiglu} Shazeer. arXiv:2002.05202.
\bibitem{geglu} Hendrycks \& Gimpel. arXiv:1606.08415.
\bibitem{poly_theory} Livni et al. AISTATS 2014.
\bibitem{adaptive} Agostinelli et al. ICLR 2015.
\bibitem{multibranch} Xu et al. NeurIPS 2021.
\bibitem{adaptive_gate} Liu et al. ICML 2022.
\bibitem{poly1} Chrysos et al. CVPR 2020.
\bibitem{poly2} Fan et al. ICLR 2021.
\bibitem{dyn_poly} Chen et al. NeurIPS 2022.
\bibitem{gelu} Hendrycks \& Gimpel. arXiv:1606.08415.
\bibitem{relu} Agarap. arXiv:1803.08375.
\bibitem{smooth} Elfwing et al. Neural Networks 2018.
\end{thebibliography}

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