The Role of Neural Network Activation Functions

TL;DR

This paper offers new theoretical insights into neural network activation functions, especially ReLU and its variants, explaining their effectiveness and the impact of regularization and architecture choices through spline theory.

Contribution

It provides a novel theoretical framework linking activation functions to spline theory, supporting design choices like regularization and skip connections in neural networks.

Findings

ReLU and variants relate to spline solutions in neural networks

Regularization strategies are justified through spline theory

Skip connections are supported by the new theoretical insights

Abstract

A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as "weight decay" and "path-norm" regularization, and provide a new justification for the use of "skip connections" in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function.