SG-Blend: Learning an Interpolation Between Improved Swish and GELU for Robust Neural Representations

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

The choice of activation function plays a crucial role in the optimization and performance of deep neural networks. While the Rectified Linear Unit (ReLU) remains the dominant choice due to its simplicity and effectiveness, its lack of smoothness may hinder gradient-based optimization in deep architectures. In this work we propose a family of $C^{2N}$-smooth activation functions whose gate follows a log-logistic CDF, achieving ReLU-like performance with purely rational arithmetic. We introduce three variants: GEM (the base family), E-GEM (an $ε$-parameterized generalization enabling arbitrary $L^p$-approximation of ReLU), and SE-GEM (a piecewise variant eliminating dead neurons with $C^{2N}$ junction smoothness). An $N$-ablation study establishes $N=1$ as optimal for standard-depth networks, reducing the GELU deficit on CIFAR-100 + ResNet-56 from 6.10% to 2.12%. The smoothness parameter $N$ further reveals a CNN-transformer tradeoff: $N=1$ is preferred for deep CNNs, while $N=2$ is preferred for transformers. On MNIST, E-GEM ties the best baseline (99.23%). On CIFAR-10 + ResNet-56, SE-GEM ($ε=10^{-4}$) surpasses GELU (92.51% vs 92.44%) -- the first GEM-family activation to outperform GELU. On CIFAR-100 + ResNet-56, E-GEM reduces the GELU deficit from 6.10% (GEM $N=2$) to just 0.62%. On GPT-2 (124M), GEM achieves the lowest perplexity (72.57 vs 73.76 for GELU), with GEM $N=1$ also beating GELU (73.32). On BERT-small, E-GEM ($ε=10$) achieves the best validation loss (6.656) across all activations. The $ε$-parameterization reveals a scale-dependent optimum: small $ε$ ($10^{-4}$--$10^{-6}$) for deep CNNs and larger transformers, with the special case of small transformers (BERT-small) benefiting from large $ε$ ($ε=10$) due to its limited depth and unconstrained gradients.

The design of modern neural architectures has converged through incremental empirical choices, yet the mechanisms governing their training dynamics remain only partially understood. We identify and analyze a negative weight drift induced by the interaction between standard losses and positively biased activation functions. We prove that under MSE or cross-entropy loss, the gradient with respect to positive pre-activations is non-negative in expectation at initialization, driving downstream weights toward negative values during early training. The drift is intrinsic to optimization rather than data, and persists across architectures (MLP, ResNet, ViT, GPT-nano, MP-SENe) and asymmetric activation functions (ReLU, GELU, SiLU). Coupled with ReLU, weight drift produces activation sparsity reaching up to 90% in GPT-nano. We characterize the sparsity-accuracy tradeoff across 79 configurations and identify a sharp accuracy cliff above $\sim$70% activation sparsity. While ReLU$^2$ achieves a good sparsity--accuracy ratio in GPT-nano, it pathologically amplifies identified activation spikes in intermediate transformer layers. Clipping resolves this while preserving the representational benefits of squaring: clipped ReLU$^2$ outperforms its unclipped version, and GELU$^2$ achieves the lowest validation loss on GPT-nano. Code is available at https://github.com/On-Point-RND/BugOrFeature.