A Structural Interpretation of GELU and Threshold-Transmission Activations via the First-Order Loss Function

Activation functions are considered an essential primitive for neural nonlinearity, i.e., they enable neural networks to serve as universal approximators. In this paper, we show that this nonlinearity can also be achieved by input-conditioned threshold gating through branches as a universal primitive. We demonstrate that standard activations -- whether piecewise-linear (ReLU, PReLU, Hardtanh) or smooth (SiLU, Sigmoid, Tanh, GELU) -- are in fact instances of a single Threshold Gating (TG) primitive. For softmax, we show that it admits an exact TG conversion via its equivalent per-element Sigmoid form. We then validate these equivalences by converting pretrained networks across CNNs, transformer-based models, and recurrent architectures, preserving model performance without requiring retraining. Threshold Gating also enables training from scratch that goes beyond replacing existing activations, enabling gains in model compression, performance, and shorter training. We also propose a 'Minimal Branch Theorem' which relates the minimum number of required branches in our primitive to the trainability of general deep neural networks. In terms of hardware implementation, TG maps to a unified implementation in the case of analog in-memory systems, addressing the bottleneck of analog-to-digital and digital-to-analog converters (ADC/DAC) that is known to significantly impact power consumption and on-chip area.

Gated Linear Units (GLU) and their variants are widely adopted in modern open-source large language model architectures and consistently outperform their non-gated counterparts, yet the underlying reasons for this advantage remain unclear. In this work, we study GLU by analyzing two-layer networks in the neural tangent kernel (NTK) regime. Our analysis reveals that the GLU structure reshapes the NTK spectrum, leading to a smaller condition number and a more compact eigenvalue distribution. Building on this finding, we further analyze the resulting training dynamics and show how the reshaped spectrum leads to faster convergence of GLU models, including a characteristic loss-crossing phenomenon observed between GLU and non-GLU models. Finally, we empirically observe that GLU has limited impact in reducing the generalization gap on various models, including ViT and GPT-2, suggesting that its primary benefit lies in accelerating optimization rather than reducing the generalization gap. The code is available at: https://github.com/Zemdalk/GLU-NTK.

Prevailing activation functions such as Swish and GELU tend toward domain-specific optima, Swish was discovered via neural architecture search on vision benchmarks, while GELU dominates transformer-based language models, and neither offers any mechanism to adapt its gating shape to individual layers. This rigidity is especially consequential in transformer FFN blocks, where LayerNorm, unlike BatchNorm, does not suppress the gradient pathologies that activation choice induces across depth. We propose SG-Blend, a per layer adaptive activation that combines SSwish, a bias-corrected, parametric Swish variant we also introduce, with learnable sharpness \b{eta} and zero-centering bias γ, with GELU through a per-layer blend coefficient α, letting each layer locate its own optimum along the SSwishGELU continuum at a cost of only three additional scalars per FFN block, with \b{eta} initialized to 1.0 and learned freely via backpropagation. On BERT-style IMDB classification (5 seeds), it matches peak accuracy (81.31%) while reducing seed-to-seed variance by 42% relative to GELU. Furthermore, it generalizes to autoregressive pretraining, achieving the lowest validation perplexity (49.10) on WikiText103 among all baselines. Crucially, ablations confirm the interpolation structure itself drives these gains, delivering reliable, top-tier performance. Beyond natural language processing, we demonstrate that SG-Blend generalizes robustly to a wider variety of tasks, extending its efficacy to computer vision and other diverse domains.