Backpropagating dE/dy by Geoffrey Hinton

1. Convert the disprepancy between each output and its target value into an error derivative.

E = 1 / 2 Sigma (Tj - Yj)^2

j in Output

dE / dYj = - (Tj -Yj)

2. Compute an error derivative in each hidden layer from error derivatives in the layer above.

dE / dZj = dYj / dZj * (dE / dYj)

, where Zj is the sum of all outputs of i hidden units.

, where Yi is the output of i hidden unit.

, where Yj is the output of j unit

dE / dZj = Yj (1 - Yj) * (dE / dYj)

, where Yj (1 - Yj) is dY / dZ of a nonlinear logic unit of y = 1 / (1 + e ^ -Z)

, where dY / dZ is y (1 - y)

dE / dYi = Sigma(j) ( dZj / dYi ) * (dE / dZj)

dE / dYi = Sigma(j) Wij * (dE / dZj)

,where dE / dZj is already computed in above layer.

thus,

dE / dWij = (dZj / dWij) * (dE / dZj)

dE / dWij = Yi * (dE / dZj)

Proof:

y = 1 / (1 + e^-Z) = (1 + e^-Z)^-1

thus, dy/dz = -1 (-e^-z) / (1 + e^-z)^2

dy/dz = 1 / (1 + e^-z) * (e^-z / (1 + e^-z) ) = y ( 1 - y)

because, (e^-z) / (1 + e^-z) = ((1 + e^-z) - 1) / (1 + e^-z)

= (1 + e^-z) / (1 + e^-z) * ( -1 / ( 1 + e^-z) ) = 1 - y

Reference:

"Learning representations by back-propagating errors" by Geoffrey Hinton (October 1986)

https://www.iro.umontreal.ca/~vincentp/ift3395/lectures/backprop_old.pdf

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