Normalized Activation Function: Toward Better Convergence

Activation functions are essential for neural networks to introduce non-linearity. A great number of empirical experiments have validated various activation functions, yet theoretical research on activation functions are insufficient. In this work, we study the impact of activation functions on the variance of gradients and propose an approach to normalize activation functions to keep the variance of the gradient same for all layers so that the neural network can achieve better convergence. First, we complement the previous work on the analysis of the variance of gradients where the impact of activation functions are just considered in an idealized initial state which almost cannot be preserved during training and obtained a property that good activation functions should satisfy as possible. Second, we offer an approach to normalize activation functions and testify its effectiveness on prevalent activation functions empirically. And by observing experiments, we discover that the speed of convergence is roughly related to the property we derived in the former part. We run experiments of our normalized activation functions against common activation functions. And the result shows our approach consistently outperforms their unnormalized counterparts. For example, normalized Swish outperforms vanilla Swish by 1.2% on ResNet50 with CIFAR-100 in terms of top-1 accuracy. Our method improves the performance by simply replacing activation functions with their normalized ones in both fully-connected networks and residual networks.