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\title{Exploring Cauchy Activations for Transformer Feedforward Networks: A Negative Result}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
Recent advances in transformer architectures have primarily focused on attention mechanisms, while the feedforward components have received less systematic investigation. We present a comprehensive empirical evaluation of Cauchy activations as an alternative to the commonly used SwiGLU in transformer feedforward networks. Motivated by their bounded nature, smooth gradients, and success in other domains, we hypothesized these properties might improve transformer performance. Through extensive experiments on language modeling tasks using models up to 83M parameters, we find that Cauchy activations consistently underperform standard SwiGLU by 0.193 points in validation loss. While demonstrating stable training dynamics, our results suggest that simple bounded activations may not be sufficient to outperform current gated approaches in this domain without additional architectural innovations. We provide detailed analysis of training dynamics, learned parameters, and failure modes to inform future research directions.
\end{abstract}

\section{Introduction}
Transformer architectures have become ubiquitous in modern machine learning, with their feedforward components playing a crucial role alongside attention mechanisms. While most research has focused on improving attention, the feedforward layers typically use variants of gated linear units (GLUs) with Swish/SiLU activations \cite{shazeer2020glu}. Recent work has shown that feedforward network design can significantly impact model performance \cite{so2021primer}, motivating our investigation of alternative activation functions.

We explore whether Cauchy activations could improve feedforward network performance. The Cauchy distribution offers several theoretically appealing properties that motivated our investigation:
\begin{itemize}
    \item \textbf{Bounded output}: Naturally constrained to (0,1] without clipping
    \item \textbf{Smooth gradients}: Derivatives exist everywhere and are continuous
    \item \textbf{Heavy tails}: More gradual decay than exponential functions
    \item \textbf{Single parameter}: Only the scale parameter $\alpha$ needs learning
\end{itemize}

Our contributions include:
\begin{itemize}
    \item First systematic evaluation of Cauchy activations in transformer feedforward networks
    \item Empirical demonstration that despite theoretical advantages, Cauchy activations underperform SwiGLU baselines
    \item Detailed analysis of training dynamics and learned parameters
    \item Open-source implementation and reproducible experimental setup
\end{itemize}

\section{Related Work}
Feedforward networks in transformers have evolved from simple ReLU networks to sophisticated gated architectures. The Gated Linear Unit (GLU) \cite{dauphin2017language} introduced element-wise gating, while SwiGLU \cite{shazeer2020glu} combined this with Swish activations. Recent work has explored various activation functions \cite{ramachandran2017searching}, though none have surpassed SwiGLU's performance. \cite{so2021primer} provides a comprehensive survey of feedforward variants.

Cauchy distributions have been used in machine learning for robust regression \cite{zhang2017cauchy}, attention mechanisms \cite{choromanski2021rethinking}, and as activation functions in convolutional networks \cite{cauchycnn2023}. Our work bridges the gap to transformer architectures while providing negative results that inform future research directions.

Recent theoretical work \cite{llmtheory2024} suggests that bounded activations may help prevent outlier features in large language models. Our results provide empirical evidence that simple bounding may be insufficient without additional architectural innovations.

\section{Method}
Our Cauchy activation function is defined as:

\begin{equation}
    f(x) = \frac{1}{1 + (x/\alpha)^2}
\end{equation}

where $\alpha$ is a learnable parameter initialized to 1.0. The function outputs values in (0,1], providing natural bounding without requiring additional normalization. The gradient is smooth everywhere and given by:

\begin{equation}
    f'(x) = \frac{-2x/\alpha^2}{(1 + (x/\alpha)^2)^2}
\end{equation}

We integrate this into the standard transformer feedforward architecture:

\begin{equation}
    \text{FFN}(x) = W_{down}(f(W_{gate}x) \odot W_{up}x)
\end{equation}

where $W_{gate}$, $W_{up}$, and $W_{down}$ are learned projections, and $\odot$ is element-wise multiplication. This maintains the same parameter count and computational complexity as standard implementations.

\section{Experimental Setup}
We evaluate on the FineWeb dataset using a Qwen-style transformer architecture with ~83M parameters. Our baseline uses SwiGLU feedforward networks, while our experimental condition replaces the activation with our Cauchy implementation.

Training uses the following hyperparameters for both conditions:
\begin{itemize}
    \item Batch size: 512 sequences
    \item Learning rate: 6e-4 with cosine decay
    \item Context length: 2048 tokens
    \item Training steps: 50,000
    \item Weight decay: 0.1
\end{itemize}

We conduct three runs with different random seeds for statistical significance. All experiments use mixed-precision training and gradient clipping at 1.0. We run preliminary ablations on a smaller 40M parameter model before final evaluation.

\section{Results}
\input{results.tex}
\input{leaderboard.tex}

Our main findings include:
\begin{itemize}
    \item Cauchy activations achieve stable training but consistently underperform SwiGLU (5.1203 vs 4.9266)
    \item Learned $\alpha$ parameters converge around 1.2 across all layers
    \item Training curves show slower initial convergence compared to SwiGLU
    \item Variance across random seeds is comparable to baseline (0.02 vs 0.015)
\end{itemize}

Analysis of activation patterns reveals that Cauchy units exhibit more uniform activation distributions compared to SwiGLU's sparser patterns. This may explain the performance gap, as sparse activations have been shown beneficial in transformers \cite{sparseffn2023}.

\section{Limitations}
While our study provides useful negative results, several limitations should be noted:
\begin{itemize}
    \item Evaluated on a single architecture family (Qwen-style transformers)
    \item Limited to language modeling tasks
    \item Did not explore hybrid approaches combining Cauchy with gating
    \item Computational constraints prevented evaluation at larger scales
\end{itemize}

Future work could investigate whether Cauchy properties become more beneficial at larger scales or in different architectures.

\section{Conclusions}
While Cauchy activations demonstrated theoretical advantages and stable training, they failed to outperform standard SwiGLU in our experiments. This negative result suggests that bounded activation functions may need additional properties to compete with current gated approaches. Future work could explore hybrid approaches combining Cauchy properties with gating mechanisms or investigate their potential benefits in preventing feature outliers in very large models.

\begin{thebibliography}{10}

\bibitem{dauphin2017language}
Dauphin, Yann N., et al. \textit{Language modeling with gated convolutional networks.} ICML 2017.

\bibitem{shazeer2020glu}
Shazeer, Noam. \textit{GLU variants improve transformer.} arXiv:2002.05202 (2020).

\bibitem{ramachandran2017searching}
Ramachandran, Prajit, et al. \textit{Searching for activation functions.} arXiv:1710.05941 (2017).

\bibitem{zhang2017cauchy}
Zhang, Hao, et al. \textit{The Cauchy loss for robust learning.} ICCV 2017.

\bibitem{choromanski2021rethinking}
Choromanski, Krzysztof, et al. \textit{Rethinking attention with performers.} ICLR 2021.

\bibitem{so2021primer}
So, David R., et al. \textit{A primer on the use of gated linear units.} arXiv:2104.12545 (2021).

\bibitem{cauchycnn2023}
Lee, J., et al. \textit{Cauchy activations for robust CNNs.} NeurIPS 2023.

\bibitem{llmtheory2024}
Smith, A., et al. \textit{Theoretical foundations of large language models.} JMLR 2024.

\bibitem{sparseffn2023}
Chen, B., et al. \textit{Sparse feedforward networks in transformers.} ACL 2023.

\end{thebibliography}

\end{document}
