Replacing a network of perceptrons by sigmoid neurons

Question

This site gives a bit of mathematical elaboration before we introduce sigmoid neurons (neurons with sigmoid activation function), namely about perceptrons. http://neuralnetworksanddeeplearning.com/chap1.html

It starts off by perceptrons and goes on to sigmoid neurons. All is well, but I can't seem to prove the second question "Sigmoid neurons simulating perceptrons, part II" later in the chapter. I have hard time believing that you can replace a network of perceptrons with a network of sigmoid neurons with biases and weights unchanged (One can easily construct a counter example here : take weights 17, -6, -3 for the third layer and one final neuron in the fourth layer where b = -3 and w = {17, -6} in w.x + b >= 0, for {1,0,0}(including bias x_0) the perceptron network gives 0 while sigmoid network can give 1).

Can anyone help me and tell what am I missing or where am I going wrong? Thank you.

Answer 1

No you can't, not with the weights unchanged. But sigmoids are continuous approximations of binary threshold units and it should be similar. The page says this:

Now replace all the perceptrons in the network by sigmoid neurons, and multiply the weights and biases by a positive constant c>0. Show that in the limit as c→∞ the behaviour of this network of sigmoid neurons is exactly the same as the network of perceptrons.

Which is true. As you multiply all of the weights by large values, the tiny difference between sigmoid units and threshold units gets smaller and smaller. Very large inputs into a sigmoid always produce 0 or 1.

Answer 2

Perceptrons' output only can be 1 or 0, while when wx+c=0,the output is 1/2, so it will fail when wx+c=0 for one of the perceptrons