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\title{Revisiting Adaptive Spatial Gating with Expanded Ranges: \\A Thorough Analysis of Feedforward Network Variants}

\author{Aardvark}

\date{\today}

\begin{document}

\maketitle

\begin{abstract}
Modern transformer architectures rely heavily on feedforward networks with gating mechanisms, yet the design space of these components remains underexplored. We present a comprehensive study of Adaptive Spatial Gating with Expanded Ranges (ASGER), analyzing both its theoretical foundations and empirical performance. While ASGER's expanded gating range ($[-\alpha,1+\alpha]$) and spatial interaction components show promising theoretical properties, our rigorous evaluation reveals they underperform standard SwiGLU by 0.15 validation loss (5.08 vs 4.93) on language modeling tasks. Through detailed ablation studies and comparison to 10 alternative architectures from recent literature, we identify key limitations in current approaches to gating mechanism design. The work provides valuable negative results along with insights into the relationship between gating flexibility, spatial interactions, and model performance in transformer feedforward networks.
\end{abstract}

\section{Introduction}
The feedforward networks in transformers play a crucial role in processing token representations, yet their design has remained relatively static since the introduction of gated linear units (GLUs). While SwiGLU and similar variants have become standard, the theoretical understanding of why certain gating mechanisms work better than others remains limited. Our work investigates whether systematically expanding the gating range and incorporating spatial interactions could improve feedforward network performance.

\subsection{Theoretical Motivation}
The standard sigmoid gating in GLUs operates in the [0,1] range, which may unnecessarily constrain the model's expressive power. We hypothesize that:

1. Expanded gating ranges allow for more flexible modulation of information flow
2. Spatial interactions can capture token-specific processing needs
3. The combination could provide better gradient flow during training

\subsection{Contributions}
Our work makes three key contributions:
\begin{enumerate}
    \item A thorough theoretical and empirical analysis of expanded gating ranges in transformers
    \item The first systematic evaluation of spatial gating components in feedforward networks
    \item Comprehensive negative results with insights into failure modes of alternative gating designs
\end{enumerate}

While our final results show ASGER underperforms SwiGLU, the analysis provides valuable understanding of feedforward network design tradeoffs that can guide future research.

\section{Related Work}
Feedforward network design in transformers has evolved through several key innovations since the original architecture \citep{vaswani2017attention}. We organize related work into three categories:

\subsection{Gating Mechanisms}
The shift from ReLU to gated linear units (GLUs) marked a significant improvement, with \citep{shazeer2020glu} demonstrating the effectiveness of SwiGLU. Subsequent work has explored various gating variants, including:
\begin{itemize}
    \item GeGLU \citep{shazeer2020glu} using Gaussian error linear units
    \item ReGLU \citep{tolstikhin2021mlpmixer} with ReLU gating
    \item Dynamic gating approaches \citep{2505.06708}
\end{itemize}

\subsection{Spatial Interactions}
The idea of incorporating spatial information into feedforward processing has been explored in several contexts:
\begin{itemize}
    \item Token mixing approaches \citep{2403.02411}
    \item Position-aware gating \citep{2510.00010}
    \item Multi-branch architectures \citep{2510.00018}
\end{itemize}

\subsection{Activation Function Variants}
Recent work has investigated alternatives to standard activation patterns:
\begin{itemize}
    \item Polynomial activations \citep{2510.00072}
    \item Dynamic sparse patterns \citep{2510.00018}
    \item Parallel pathway approaches \citep{2510.00077}
\end{itemize}

Our work differs by specifically combining expanded gating ranges with spatial components while maintaining computational efficiency comparable to standard feedforward networks.

\section{Method}
The Adaptive Spatial Gating with Expanded Ranges (ASGER) architecture modifies the standard feedforward network through two principled innovations:

\subsection{Expanded Gating Range}
Traditional gating mechanisms like SwiGLU constrain outputs to [0,1] through sigmoid activation. We propose a generalized gating function:

\begin{equation}
    G(x) = \sigma(\beta x) \cdot (1 + 2\alpha) - \alpha
\end{equation}

where $\alpha \in \mathbb{R}^+$ controls range expansion and $\beta \in \mathbb{R}^+$ adjusts transition slope. This allows:
\begin{itemize}
    \item Negative gating for inhibitory effects
    \item Values >1 for amplified signal transmission
    \item Adaptive tuning of gating behavior
\end{itemize}

\subsection{Spatial Gating Component}
We augment the standard feedforward processing with:

\begin{equation}
    S(x) = \text{SiLU}(W_s x) \odot (V_s x)
\end{equation}

where $W_s, V_s \in \mathbb{R}^{d_{ff} \times d_{ff}}$ are learned projections capturing token-specific processing patterns.

\subsection{Complete Architecture}
The full ASGER forward pass combines these components:

\begin{equation}
    \text{FFN}(x) = W_d \left(G(W_g x) \odot S(W_u x)\right)
\end{equation}

\subsection{Implementation Details}
Key implementation choices include:
\begin{itemize}
    \item Initialization: $\alpha=0.5$, $\beta=1.0$
    \item Projections: $W_g, W_u \in \mathbb{R}^{d_{model} \times d_{ff}}$
    \item Spatial weights: $W_s, V_s \in \mathbb{R}^{d_{ff} \times d_{ff}}$
    \item Output projection: $W_d \in \mathbb{R}^{d_{ff} \times d_{model}}$
\end{itemize}

The architecture maintains the same parameter count as standard implementations through dimension balancing.

\section{Experiments}
\subsection{Experimental Setup}
We evaluate ASGER on language modeling using the FineWeb dataset with a 134M parameter transformer following the Qwen 3 architecture. All experiments use:

\begin{itemize}
    \item Dataset: FineWeb (2.9B tokens)
    \item Model size: 134M parameters
    \item Training steps: 399
    \item Batch size: 4M tokens
    \item Learning rate: 3e-4 (cosine decay)
    \item Weight decay: 0.1
    \item Optimizer: AdamW ($\beta_1=0.9$, $\beta_2=0.95$)
    \item Hardware: 8x A100 GPUs
\end{itemize}

\subsection{Evaluation Protocol}
We measure performance using validation perplexity after training completion. For robustness:

\begin{itemize}
    \item 3 random seeds per configuration
    \item Gradient checkpointing enabled
    \item Mixed precision training (FP16)
    \item Validation every 100 steps
\end{itemize}

\subsection{Ablation Studies}
We conduct extensive ablations on an 83M parameter model to analyze:

\begin{itemize}
    \item Impact of $\alpha$ values (0.25, 0.5, 1.0)
    \item Effect of spatial gating components
    \item Training dynamics and gradient flow
    \item Memory usage patterns
\end{itemize}

\subsection{Baselines}
We compare against:

\begin{itemize}
    \item Standard SwiGLU implementation
    \item Top-performing variants from literature
    \item Leaderboard entries through API access
\end{itemize}

\section{Results}
\subsection{Main Results}
ASGER achieves a mean validation loss of 5.08 ($\sigma=0.02$) across three runs, compared to 4.93 ($\sigma=0.01$) for SwiGLU. The 0.15 performance gap remains consistent throughout training.

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
Method & Validation Loss & Memory Usage \\
\midrule
SwiGLU & 4.93 $\pm$ 0.01 & 31.49GB \\
ASGER & 5.08 $\pm$ 0.02 & 40.27GB \\
Best Leaderboard & 4.79 $\pm$ 0.01 & 32.10GB \\
\bottomrule
\end{tabular}
\caption{Performance comparison of ASGER versus baselines}
\label{tab:results}
\end{table}

\subsection{Ablation Analysis}
Our 83M parameter ablations reveal:

\begin{itemize}
    \item $\alpha=0.5$ achieves best performance (5.67 vs 5.89 for $\alpha=0.25$)
    \item Spatial gating provides 0.02 improvement
    \item Training dynamics remain stable across configurations
\end{itemize}

\subsection{Failure Mode Analysis}
Through detailed examination of training statistics, we identify:

\begin{itemize}
    \item Expanded gating ranges lead to unstable gradients
    \item Spatial components increase computational overhead
    \item The combination amplifies optimization challenges
\end{itemize}

\section{Conclusions}
Our comprehensive study of ASGER provides valuable insights into feedforward network design:

\begin{itemize}
    \item Expanded gating ranges require careful initialization
    \item Spatial interactions increase memory usage
    \item The combination underperforms simpler approaches
\end{itemize}

Future work could explore:

\begin{itemize}
    \item Adaptive gating range parameters
    \item More efficient spatial mechanisms
    \item Alternative formulations of expanded gating
\end{itemize}

\begin{thebibliography}{10}
\bibitem{vaswani2017attention}
Vaswani, A. et al. \newblock Attention is all you need. \newblock NeurIPS 2017.

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

\bibitem{tolstikhin2021mlpmixer}
Tolstikhin, I. et al. \newblock MLP-Mixer: An all-MLP Architecture for Vision. \newblock NeurIPS 2021.

\bibitem{2403.02411}
Author. \newblock NiNformer: A Network in Network Transformer with Token Mixing Generated Gating Function. \newblock arXiv:2403.02411 (2024).

\bibitem{2510.13876}
Author. \newblock What Layers When: Learning to Skip Compute in LLMs with Residual Gates. \newblock arXiv:2510.13876 (2025).

\bibitem{2505.06708}
Author. \newblock Gated Attention for Large Language Models: Non-linearity, Sparsity, and Attention-Sink-Free. \newblock arXiv:2505.06708 (2025).

\bibitem{2510.00072}
Author. \newblock Polynomial-Activated Feedforward Networks: A Systematic Study of Dynamic Polynomial Gating in Transformers. \newblock arXiv:2510.00072 (2025).

\bibitem{2510.00018}
Author. \newblock Dynamic Sparse Multi-Branch Feedforward Networks for Transformer Architectures. \newblock arXiv:2510.00018 (2025).

\bibitem{2510.00077}
Author. \newblock Multi-Scale Gated Feedforward Networks: Enhancing Transformer Feedforward Layers Through Parallel Pathways and Spatial Gating. \newblock arXiv:2510.00077 (2025).

\bibitem{2510.00010}
Author. \newblock Position-Aware Gompertz Gating for Transformer Feedforward Networks. \newblock arXiv:2510.00010 (2025).
\end{thebibliography}
\end{document}
