In this problem, we will walk through a single step of the gradient descent algorithm for logistic

regression. Assume two dimension input. Recap:

f $($x; w$,$ b$)=\backslash$sigma $($w $$ x $+$ b$)$

Cross entropy loss L$($y$,$ y$)=[$y log $$y $+(1$ y$)$ log$(1$y$)]$

The single update step $\backslash$theta t$+1=\backslash$theta t $\backslash$eta $\backslash$theta L$($f $($x; $\backslash$theta $),$ y$),$ where $\backslash$theta $=[$w$1,$ w$2,$ b$]$T

Now given

Initial parameters : w$1=$ w$2=$ b $=0,(\backslash$theta $0=[0,0,0]))$

Learning rate $\backslash$eta $=0\mathrm{.}1$

data example : x $=[3,2],$ y $=1$

$($a$)(4$ pts$)$ Compute the first gradient $\backslash$theta L$($f $($x; $\backslash$theta $),$ y$).$

$($b$)(4$ pts$)$ Compute the updated parameter vector $\backslash$theta $1$ from the single update step.