Consider a specific $2$ hidden layer ReLU network with inputs xinR,$1$

dimensional outputs, and $2$ neurons per hidden layer. This function is

given by

$h(x)=W^{(3)}$max$\{0,W^{(2)}$max$\{0,W^{(1)}x+$vec$(b{)}^{(1)}\}+b^{(2)}\}+b^{(3)}$

where the max is element$-$wise, with weights:

$W^{(1)}=\left[\begin{array}{c}1\mathrm{.}5\end{array}\right]$

$0\mathrm{.}5$

$b^{(1)}=\left[\begin{array}{c}0\end{array}\right]$

$1$

$W^{(2)}=\left[\begin{array}{ccc}1& 2& 1\end{array}\right]$

$2$

$b^{(2)}=\left[\begin{array}{c}0\end{array}\right]$

$1$

$W^{(3)}=\left[\begin{array}{cc}1& 1\end{array}\right]$

$b^{(3)}=-1$

An interesting property of networks with piece$-$wise linear activations

like the ReLU is that on the whole they compute piece$-$wise linear

functions. At each of the following points $x=x_o,$ determine the value

of the new weight WinR and bias binR such that $\frac{dh(x)}{dx}{|}_x=x_0=W$ and

$W{x}_o+b=h(x_o).$

$x_o=2$

$x_o=-1$

$x_o=1$