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 $\boldsymbol{\Theta}=[w_1,w_2,b_1,b_2]$.

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