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

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