In back propagation for neural networks, what exactly is the "error signal"? For example:
Imagine we end up with a sum of 0.755 on our output node.
We then apply the activation function (in this case I'll use a sigmoid) which gives us a final value of 0.68.
Now imagine the actual output we were looking for was 0. This means our error was 0 - 0.68 or -0.68.
Now, this is the weird part... all the literature then says we need to calculate the "error signal" by multiplying the derivative of the sigmoid at 0.755 by this error.
ie. Error Signal = S'(0.755) * (0 - 0.68)
My question is, what exactly is this 'Error signal'? I had thought maybe it was something like Euler's Method but we're multiplying by the error in y here rather than a change in x so that can't be right.
 A: Back propagation training uses gradient descent.  The most common error function used for the gradient descent is the sum of the square errors of the network/layer outputs: $$ e(i)=\frac{1}{2}\sum_i(desired_i - actual_i)^2.$$  To update each weight we need the partial derivative of the error function with respect to the weight being updated.  Using the chain rule, this partial derivative equals $$(actual - desired)*e'(input).$$  For a fuller derivation look here.  
So the simple answer to your question is that error signal is the gradient of the error function at the input level.  
A: The output value of a node depends on the inputs from several nodes. The relative contribution of each of those input nodes determines what proportion of the total error each input node is responsible for. So multiplying the error by the node output gives a signal showing how much that node needs to change to reduce the error in the most efficient way possible. This is an example of the credit assignment problem: given an error, how do we decide which weights in the model are responsible and how to optimise those weights?
