Problem $1$ For multi$-$layer perceptron,

s$($l$)=$ W $($l$)$h$($l$1)+$ b$($l$),$ h$($l$)=$ $($s$($l$)),$ l $=1,...,$ L$,$

where s $=$ sL $=$ hL is the top layer logit score for logistic regression, h$0=$ x is the input, and $()$

is element$-$wise ReLU.

$(1)$ Derive $$ log p$($y$|$s$)/$s at the top layer.

$(2)$ Derive the back$-$propagation algorithm. You can first derive results in terms of elements of

matrices and vectors, and then arrange the results in terms of matrices and vectors.