A General Framework for Activation Function Optimization Based on Mollification Theory

The deep learning paradigm is progressively shifting from non-smooth activation functions, exemplified by ReLU, to smoother alternatives such as GELU and SiLU. This transition is motivated by the fact that non-differentiability introduces challenges for gradient-based optimization, while an expanding body of research demonstrates that smooth activations yield superior convergence, improved generalization, and enhanced training stability. A central challenge, however, is how to systematically transform widely used non-smooth functions into smooth counterparts that preserve their proven representational strengths while improving differentiability and computational efficiency. To address this, we propose a general activation smoothing framework grounded in mollification theory. Leveraging the Epanechnikov kernel, the framework achieves statistical optimality and computational tractability, thereby combining theoretical rigor with practical utility. Within this framework, we introduce Smoothed ReLU (S-ReLU), a novel second-order continuously differentiable (C2) activation derived from ReLU that inherits its favorable properties while mitigating inherent drawbacks. Extensive experiments on CIFAR-10, CIFAR-100, and ImageNet-1K with Vision Transformers and ConvNeXt consistently demonstrate the superior performance of S-ReLU over existing ReLU variants. Beyond computer vision, large-scale fine-tuning experiments on language models further show that S-ReLU surpasses GELU, underscoring its broad applicability across both vision and language domains and its potential to enhance stability and scalability.