On Approximation Capabilities of ReLU Activation and Softmax Output Layer in Neural Networks

TL;DR

This paper extends universal approximation theory to neural networks with ReLU and softmax layers, proving their capacity to approximate any function or indicator function in $L^1$, thus providing a theoretical basis for softmax in classification.

Contribution

It is the first to theoretically justify the use of softmax output layers in neural networks for pattern classification tasks.

Findings

ReLU networks can approximate any $L^1$ function with arbitrary precision.

Softmax output layers can approximate indicator functions in $L^1$, suitable for classification.

First theoretical validation of softmax layers in neural network approximation capabilities.

Abstract

In this paper, we have extended the well-established universal approximator theory to neural networks that use the unbounded ReLU activation function and a nonlinear softmax output layer. We have proved that a sufficiently large neural network using the ReLU activation function can approximate any function in $L^1$ up to any arbitrary precision. Moreover, our theoretical results have shown that a large enough neural network using a nonlinear softmax output layer can also approximate any indicator function in $L^1$, which is equivalent to mutually-exclusive class labels in any realistic multiple-class pattern classification problems. To the best of our knowledge, this work is the first theoretical justification for using the softmax output layers in neural networks for pattern classification.