Why must a nonlinear activation function be used i

Question

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?

Answer 1

If we only allow linear activation functions in a neural network, the output will just be a linear transformation of the input, which is not enough to form a universal function approximator. Such a network can just be represented as a matrix multiplication, and you would not be able to obtain very interesting behaviors from such a network.

The same thing goes for the case where all neurons have affine activation functions (i.e. an activation function on the form f(x) = a*x + c, where a and c are constants, which is a generalization of linear activation functions), which will just result in an affine transformation from input to output, which is not very exciting either.

A neural network may very well contain neurons with linear activation functions, such as in the output layer, but these require the company of neurons with a non-linear activation function in other parts of the network.

Note: An interesting exception is DeepMind's synthetic gradients, for which they use a small neural network to predict the gradient in the backpropagation pass given the activation values, and they find that they can get away with using a neural network with no hidden layers and with only linear activations.

Answer 2

Several good answers are here. It will be good to point out the book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. It is a book worth referring to for getting a deeper insight about several ML related concepts. Excerpt from page 229 (section 5.1):

If the activation functions of all the hidden units in a network are taken to be linear, then for any such network we can always find an equivalent network without hidden units. This follows from the fact that the composition of successive linear transformations is itself a linear transformation. However, if the number of hidden units is smaller than either the number of input or output units, then the transformations that the network can generate are not the most general possible linear transformations from inputs to outputs because information is lost in the dimensionality reduction at the hidden units. In Section 12.4.2, we show that networks of linear units give rise to principal component analysis. In general, however, there is little interest in multilayer networks of linear units.

Answer 3

There are times when a purely linear network can give useful results. Say we have a network of three layers with shapes (3,2,3). By limiting the middle layer to only two dimensions, we get a result that is the "plane of best fit" in the original three dimensional space.

But there are easier ways to find linear transformations of this form, such as NMF, PCA etc. However, this is a case where a multi-layered network does NOT behave the same way as a single layer perceptron.

Answer 4

A layered NN of several neurons can be used to learn linearly inseparable problems. For example XOR function can be obtained with two layers with step activation function.

Answer 5

"The present paper makes use of the Stone-Weierstrass Theorem and the cosine squasher of Gallant and White to establish that standard multilayer feedforward network architectures using abritrary squashing functions can approximate virtually any function of interest to any desired degree of accuracy, provided sufficently many hidden units are available." (Hornik et al., 1989, Neural Networks)

A squashing function is for example a nonlinear activation function that maps to [0,1] like the sigmoid activation function.

Answer 6

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: