Adding a hidden layer

We change our simple model to (see graph) a network with just one hidden layer but with scalar variables only.

Our output variable changes to \( a_2 \) and \( a_1 \) is now the output from the hidden node and \( a_0=x \). We have then

$$ z_1 = w_1a_0+b_1 \hspace{0.1cm} \wedge a_1 = \sigma_1(z_1), $$ $$ z_2 = w_2a_1+b_2 \hspace{0.1cm} \wedge a_2 = \sigma_2(z_2), $$

and the cost function

$$ C(x;\boldsymbol{\Theta})=\frac{1}{2}(a_2-y)^2, $$

with \( \boldsymbol{\Theta}=[w_1,w_2,b_1,b_2] \).