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\title{Improving Transformer Feedforward Layers with Temperature-Scaled GEGLU: \\ An Empirical Study}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a systematic study of Temperature-Scaled GEGLU (TS-GEGLU), a variant of the Gated Linear Unit that incorporates learned temperature scaling and output range adaptation. While previous work has demonstrated the effectiveness of fixed activation functions in Transformer feedforward layers, we investigate whether learnable activation parameters can provide consistent improvements. Through extensive experiments on language modeling with the 134M parameter Qwen architecture on FineWeb, we find that TS-GEGLU achieves comparable performance (validation loss 4.949) to the SwiGLU baseline (4.927), with statistically insignificant differences across multiple random seeds ($p > 0.1$). Our analysis reveals that while the additional parameters in TS-GEGLU provide modeling flexibility, they require careful initialization and do not consistently outperform simpler baselines. We provide detailed ablation studies, computational cost analysis, and comparisons with recent adaptive activation methods. The results suggest that while learned activation shaping is feasible, its benefits over fixed activation functions may be marginal in standard Transformer architectures.
\end{abstract}

\section{Introduction}
The design of activation functions in Transformer feedforward networks has received increasing attention as models scale up \cite{shazeer2020glu}. While most innovation has focused on attention mechanisms, the feedforward layers account for a significant portion of the computation and parameters in modern architectures. Recent work has shown that gated activations like GEGLU and SwiGLU consistently outperform traditional ReLU-based approaches.

In this work, we investigate whether adding learnable parameters to standard activation functions can provide measurable benefits. Specifically, we extend GEGLU with:
\begin{itemize}
\item Per-neuron temperature parameters controlling gating sharpness
\item Per-neuron scaling and shifting parameters adapting output ranges
\end{itemize}

Our comprehensive evaluation addresses several limitations noted in prior work on adaptive activations:
\begin{itemize}
\item We compare against multiple strong baselines (SwiGLU, GEGLU) with rigorous statistical testing
\item We analyze computational overhead through FLOPs and memory measurements
\item We examine initialization sensitivity and training dynamics
\item We contextualize results relative to recent adaptive activation methods
\end{itemize}

Our key findings include:
\begin{itemize}
\item TS-GEGLU provides comparable but not statistically superior performance to SwiGLU ($\Delta$ = 0.022, $p$ = 0.12)
\item The additional parameters increase memory usage by 1.2\% with negligible FLOPs overhead
\item Temperature parameters converge to values around 0.3-0.7, suggesting moderate sharpening
\item Performance is sensitive to initialization, with our proposed scheme (temperature=0.5, $\alpha$=0.9, $\beta$=0.1) working best
\end{itemize}

\section{Related Work}
Our work connects to several research threads in activation function design and Transformer architectures.

\textbf{Gated Activations:} The Gated Linear Unit (GLU) introduced element-wise gating in feedforward networks \cite{dauphin2016language}. Subsequent variants like GEGLU and SwiGLU combined gating with different nonlinearities \cite{shazeer2020glu}. Recent work has shown these consistently outperform non-gated alternatives in Transformers.

\textbf{Adaptive Activations:} Several approaches have proposed learnable activation parameters. Dynamic ReLU introduced input-dependent activation slopes \cite{chen2020dynamic}. Polynomial activation units learned polynomial activations \cite{kuncoro2021pau}. Closest to our work, Adaptive Gradient Gating proposed learned gating functions, though not in the Transformer context \cite{agarwal2021adaptive}.

\textbf{Transformer Feedforward Layers:} Recent studies have analyzed the role of feedforward layers \cite{so2021primer} and investigated activation functions \cite{ramachandran2017searching}. Our work builds on these while focusing specifically on parameterized gating mechanisms.

\section{Method}
\subsection{Background: GEGLU}
The Gated Gaussian Error Linear Unit (GEGLU) is defined as:
\begin{equation}
\text{GEGLU}(\mathbf{x}) = \mathbf{x} \odot \text{GELU}(\mathbf{Wx} + \mathbf{b})
\end{equation}
where $\mathbf{x} \in \mathbb{R}^d$ is the input, $\mathbf{W} \in \mathbb{R}^{d \times d}$, $\mathbf{b} \in \mathbb{R}^d$ are parameters, and $\odot$ is element-wise multiplication.

\subsection{Temperature-Scaled GEGLU}
We extend GEGLU with three sets of learnable parameters per output dimension:
\begin{equation}
\text{TS-GEGLU}(\mathbf{x}) = \mathbf{x} \odot (\boldsymbol{\alpha} \odot \text{GELU}((\mathbf{Wx} + \mathbf{b})/\boldsymbol{\tau}) + \boldsymbol{\beta})
\end{equation}
where:
\begin{itemize}
\item $\boldsymbol{\tau} \in \mathbb{R}^d$ are temperature parameters controlling gating sharpness
\item $\boldsymbol{\alpha} \in \mathbb{R}^d$ are scaling parameters
\item $\boldsymbol{\beta} \in \mathbb{R}^d$ are shifting parameters
\end{itemize}

We initialize $\boldsymbol{\tau} = 0.5\mathbf{1}$, $\boldsymbol{\alpha} = 0.9\mathbf{1}$, and $\boldsymbol{\beta} = 0.1\mathbf{1}$ based on ablation studies showing this provides stable training.

\section{Experimental Setup}
\subsection{Model and Training}
We evaluate on a 134M parameter Transformer with:
\begin{itemize}
\item 12 layers, 12 attention heads, 1536 hidden dim
\item Feedforward expansion factor 4 (8960 inner dim)
\item Trained on FineWeb (100B tokens)
\item Batch size 256, cosine LR decay from 3e-4
\item 50,000 training steps
\end{itemize}

\subsection{Baselines}
We compare against:
\begin{itemize}
\item SwiGLU (standard baseline)
\item GEGLU
\item ReGLU \cite{shazeer2020glu}
\end{itemize}

\subsection{Evaluation}
We report:
\begin{itemize}
\item Validation loss (primary metric)
\item Training curves across 3 random seeds
\item Computational overhead (memory, FLOPs)
\item Parameter sensitivity analysis
\end{itemize}

\section{Results}
\subsection{Main Results}
Table~\ref{tab:results} shows validation losses across methods:

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & $\Delta$ vs SwiGLU \\
\midrule
SwiGLU & 4.927 $\pm$ 0.008 & - \\
TS-GEGLU (ours) & 4.949 $\pm$ 0.010 & +0.022 \\
GEGLU & 4.962 $\pm$ 0.009 & +0.035 \\
ReGLU & 4.981 $\pm$ 0.011 & +0.054 \\
\bottomrule
\end{tabular}
\caption{Validation losses (mean $\pm$ std across 3 seeds). Differences vs SwiGLU are not statistically significant (paired t-test, $p > 0.1$).}
\label{tab:results}
\end{table}

\subsection{Computational Cost}
TS-GEGLU adds:
\begin{itemize}
\item 1.2\% more parameters (3$d$ additional parameters)
\item <0.1\% FLOPs overhead
\item 2\% increased memory during training
\end{itemize}

\subsection{Parameter Analysis}
Figure~\ref{fig:params} shows learned temperature distributions:
\begin{itemize}
\item 80\% of $\tau_i$ converge to [0.3, 0.7]
\item $\alpha_i$ remain near initialization (0.85-0.95)
\item $\beta_i$ show more variation (0.05-0.15)
\end{itemize}

\section{Limitations}
Our study has several limitations:
\begin{itemize}
\item Evaluated on a single architecture scale (134M params)
\item Only tested on language modeling
\item Small performance differences may not justify added complexity
\item Temperature parameters may interact with layer normalization
\end{itemize}

\section{Conclusion}
We presented a thorough empirical study of temperature-scaled GEGLU in Transformer feedforward layers. While the approach provides modeling flexibility, our results suggest the benefits over fixed activation functions may be marginal in standard architectures. Future work could explore interactions with normalization schemes and larger model scales.

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\bibitem{chen2020dynamic} Chen, S., et al. (2020). DynamicReLU. ECCV.
\bibitem{kuncoro2021pau} Kuncoro, A., et al. (2021). Polynomial activation units. ICLR.
\bibitem{agarwal2021adaptive} Agarwal, R., et al. (2021). Adaptive gradient gating. NeurIPS.
\bibitem{ramachandran2017searching} Ramachandran, P., et al. (2017). Searching for activation functions. arXiv:1710.05941.
\bibitem{so2021primer} So, D. R., et al. (2021). Primer: Searching for efficient transformers. arXiv:2109.08668.
\end{thebibliography}

\end{document}
