Why must a nonlinear activation function be used in a backpropagation neural network?

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I've been reading some things on neural networks and I understand the general principle of a single layer neural network. I understand the need for aditional layers, but why are nonlinear activation functions used?

This question is followed by this one: What is a derivative of the activation function used for in backpropagation?

Best Solution

The purpose of the activation function is to introduce non-linearity into the network

in turn, this allows you to model a response variable (aka target variable, class label, or score) that varies non-linearly with its explanatory variables

non-linear means that the output cannot be reproduced from a linear combination of the inputs (which is not the same as output that renders to a straight line--the word for this is affine).

another way to think of it: without a non-linear activation function in the network, a NN, no matter how many layers it had, would behave just like a single-layer perceptron, because summing these layers would give you just another linear function (see definition just above).

>>> in_vec = NP.random.rand(10)
>>> in_vec
  array([ 0.94,  0.61,  0.65,  0.  ,  0.77,  0.99,  0.35,  0.81,  0.46,  0.59])

>>> # common activation function, hyperbolic tangent
>>> out_vec = NP.tanh(in_vec)
>>> out_vec
 array([ 0.74,  0.54,  0.57,  0.  ,  0.65,  0.76,  0.34,  0.67,  0.43,  0.53])

A common activation function used in backprop (hyperbolic tangent) evaluated from -2 to 2:

enter image description here