Let's consider a concrete example of a two$-$layer neural network, that uses a $D-$dimensional

feature vector $x$ to predict a $K-$dimensional outcome vector of $y.$ The hidden layer consists of

$M-$dimensional vector of $z.$ The model is described with the following functions:

$z_j=h(a_j),$ with $a_j={\displaystyle \underset{i=0}{\overset{D}{}}}{w}_{ji}^{(1)}{x}_i,$

$y_k=(a_k),$ with $a_k={\displaystyle \underset{j=0}{\overset{M}{}}}{w}_{kj}^{(2)}{z}_j.$

For simplicity, we consider a linear activation function for the output units, so that $y_k=a_K($i$.$e$.$

$\{$:$(a_k)=a_k).$ We consider a logistic sigmoid activation function for the hidden units, given by