$4\mathrm{.}2(10$ points$)$ Derive Gradient

Given a training dataset Straining $=\{($xi$,$ yi$)\},$ i $=1,...,$ n$\},$ we wish to optimize the

negative log$-$likelihood loss L$($w$,$ b$)$ of the logistic regression model defined above:

n

L$($w$,$b$)=$Xlnpi $(5)$

i$=1$

where pi $=$ p$($yi$|$xi$).$ The optimal weight vector w and bias b are used to build the

logistic regression model:

w$,$b$=$argminL$($w$,$b$)(6)$ w$,$b

In this problem, we attempt to obtain the optimal parameters w$$ and b$$ by using a standard gradient descent algorithm.

$($a$)$ Please show that

$$ L $($ w $,$ b $)$w

$=$

Xn i$=1$

$(1$ pi$)$yixi$.$