Rethinking Activation Functions: Reducing Bias Can Improve Learning

This paper proposes $\mathrm{dynActivation}$, a per-layer trainable activation defined as $f_i(x) = \mathrm{BaseAct}(x)(α_i - β_i) + β_i x$, where $α_i$ and $β_i$ are lightweight learned scalars that interpolate between the base nonlinearity and a linear path and $\mathrm{BaseAct}(x)$ resembles any ReLU-like function. The static and dynamic ReLU-like variants are then compared across multiple vision tasks, language modeling tasks, and ablation studies. The results suggest that dynActivation variants tend to linearize deep layers while maintaining high performance, which can improve training efficiency by up to $+54\%$ over ReLU. On CIFAR-10, dynActivation(Mish) improves over static Mish by up to $+14.02\%$ on AttentionCNN with an average improvment by $+6.00\%$, with a $24\%$ convergence-AUC reduction relative to Mish (2120 vs. 2785). In a 1-to-75-layer MNIST depth-scaling study, dynActivation never drops below $95\%$ test accuracy ($95.3$--$99.3\%$), while ReLU collapses below $80\%$ at 25 layers. Under FGSM at $\varepsilon{=}0.08$, dynActivation(Mish) incurs a $55.39\%$ accuracy drop versus $62.79\%$ for ReLU ($7.40\%$ advantage). Transferred to language modeling, a new proposed dynActGLU-variant achieves a $10.3\%$ relative perplexity reduction over SwiGLU at 5620 steps (4.047 vs. 4.514), though the gap vanishes at 34300 steps.

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.

We analyze a simple one-hidden-layer neural network with ReLU activation functions and fixed biases, with one-dimensional input and output. We study both continuous and discrete versions of the model, and we rigorously prove the convergence of the learning process with the $L^2$ squared loss function and the gradient descent procedure. We also prove the spectral bias property for this learning process. Several conclusions of this analysis are discussed; in particular, regarding the structure and properties that activation functions should possess, as well as the relationships between the spectrum of certain operators and the learning process. Based on this, we also propose an alternative activation function, the full-wave rectified exponential function (FReX), and we discuss the convergence of the gradient descent with this alternative activation function.