The Mathematical Relationship Between Layer Normalization and Dynamic Activation Functions

A recent paper proposes Dynamic Tanh (DyT) as a drop-in replacement for layer normalization (LN). Although the method is empirically well-motivated and appealing from a practical point of view, it lacks a theoretical foundation. In this work, we shed light on the mathematical relationship between layer normalization and dynamic activation functions. In particular, we derive DyT from LN and show that a well-defined approximation is needed to do so. By dropping said approximation, an alternative activation function is obtained, which we call Dynamic Inverse Square Root Unit (DyISRU). DyISRU is the exact counterpart of layer normalization, and we demonstrate numerically that it indeed resembles LN more accurately than DyT does.