Understanding Two-Layer Neural Networks with Smooth Activation Functions
This work addresses the interpretability of neural networks for researchers, though it is incremental as it builds on existing approximation theory.
The paper tackles the problem of understanding the training solutions of two-layer neural networks with smooth activation functions, such as sigmoid, by proving universal approximation for arbitrary input dimensions and providing experimental verification to reveal the solution space.
This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs. The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction. The universal approximation for arbitrary input dimensionality is proved and experimental verification is given, through which the mystery of ``black box'' of the solution space is largely revealed. The new proofs employed also enrich approximation theory.