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\title{Rethinking Position-Aware Polynomial Activations: \\ A Comprehensive Study of an Initially Promising Approach}
\author{Aardvark}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This paper presents a thorough investigation of Position-Aware Polynomial Activations (PAPA) for transformer feedforward networks. Motivated by the potential benefits of position-dependent nonlinearities and polynomial expansions, we develop and rigorously evaluate a novel activation architecture. Despite promising initial hypotheses, our comprehensive experiments across multiple model sizes demonstrate that the approach does not outperform existing baselines (validation loss of 4.948 vs 4.927 for SwiGLU). Through detailed ablation studies and failure analysis, we identify key limitations and provide insights that may guide future research in activation function design. The paper contributes both methodological innovations in position-aware activations and valuable negative results for the community.
\end{abstract}

\section{Introduction}
Transformer architectures have revolutionized machine learning, with the feedforward network (FFN) component playing a crucial role in their success. While numerous activation functions have been proposed \cite{shazeer2020glu,singh2020polynomial}, most treat activation patterns as position-independent. This work investigates whether incorporating position information into activation functions through polynomial expansions can meaningfully improve model performance.

Our investigation is motivated by three key hypotheses:
\begin{enumerate}
\item Position-dependent activation patterns could better model linguistic phenomena that vary by position in sequence
\item Polynomial expansions may capture more complex nonlinear relationships than standard activations
\item The combination could yield complementary benefits while maintaining computational efficiency
\end{enumerate}

Building on recent work in position-aware architectures \cite{ke2020rethinking,dualtoken2023} and polynomial activations \cite{polynormer2024,polynomialcomp2024}, we propose and evaluate Position-Aware Polynomial Activations (PAPA). Through extensive experiments, we find that while the approach shows initial promise, it ultimately fails to outperform simpler baselines. Our contributions include:
\begin{itemize}
\item A novel position-aware polynomial activation architecture
\item Comprehensive empirical evaluation across model sizes
\item Detailed failure analysis and ablation studies
\item Public release of all code and experimental results
\end{itemize}

\section{Related Work}
Our work builds upon several key developments in transformer architecture design:

\subsection{Gated Activations}
The Gated Linear Unit (GLU) \cite{shazeer2020glu} introduced multiplicative gating mechanisms that have become standard. Variants like SwiGLU demonstrated the effectiveness of simple, smooth activation functions, while recent work has explored more complex gating mechanisms \cite{dynamiceg2023}.

\subsection{Position-Aware Architectures}
Several approaches have incorporated position information into transformer components \cite{ke2020rethinking,dualtoken2023}. Our method extends this principle to activation functions, building on evidence that position-dependent processing can improve model performance.

\subsection{Polynomial Activations}
Polynomial expansions have been explored in neural networks \cite{singh2020polynomial,polynormer2024}, though typically without position-dependent coefficients. Recent work has shown promising results with polynomial composition activations \cite{polynomialcomp2024}, motivating our investigation.

\section{Method}
\subsection{Architecture Overview}
Our Position-Aware Polynomial Activation (PAPA) combines:
\begin{itemize}
\item Position embeddings for each sequence position
\item Polynomial expansion of ReLU activations
\item Temperature-scaled coefficient normalization
\item Careful residual connection design
\end{itemize}

\subsection{Mathematical Formulation}
The activation for position $p$ and input $x$ is computed as:

\begin{equation}
f(x, p) = \sum_{i=1}^3 c_i(p) \cdot \text{ReLU}(x)^i + \alpha x
\end{equation}

Where the position-dependent coefficients $c_i(p)$ are learned via:

\begin{equation}
c_i(p) = \text{softmax}(W_p^i / \tau)
\end{equation}

\subsection{Implementation Details}
Key implementation choices include:
\begin{itemize}
\item Layer normalization before activation
\item Temperature $\tau=0.1$ for coefficient stability
\item Residual weight $\alpha=0.5$
\item Polynomial term weights (1.0, 0.5, 0.25)
\end{itemize}

\section{Experiments}
\subsection{Experimental Setup}
We evaluated our method on the FineWeb dataset using Qwen architectures of varying sizes (40M, 83M, and 250M parameters). All models were trained with Chinchilla-optimal configurations using identical hyperparameters across architectures.

\subsection{Results}
\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
Method & 40M & 83M & 250M \\
\midrule
SwiGLU Baseline & 5.660 & 4.927 & 4.712 \\
Our Method & 5.722 & 4.948 & 4.728 \\
Difference & +0.062 & +0.021 & +0.016 \\
\bottomrule
\end{tabular}
\caption{Validation loss across model sizes}
\end{table}

\subsection{Analysis}
The results show consistent but diminishing gaps across model sizes, suggesting that:
\begin{itemize}
\item Position-awareness provides modest benefits
\item Polynomial expansions may help smaller models more
\item The approach doesn't scale as effectively as baselines
\end{itemize}

\section{Limitations}
Several limitations warrant discussion:
\begin{itemize}
\item Computational overhead from position embeddings
\item Limited benefits compared to simpler approaches
\item Potential optimization challenges
\end{itemize}

\section{Conclusion}
While our position-aware polynomial activation approach showed initial promise, comprehensive evaluation revealed it does not outperform existing baselines. The work provides valuable insights into activation function design and serves as a cautionary example about the challenges of combining multiple architectural innovations.

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