Implementing Classification Model: First some background for classification:

$$ You are given labeled data $\{($xi$,$ yi$)\}$N

i$=1$ for xi in Rd and yi in $\{1,1\}.$

$$ Logistic regression involves choosing a label according to

y $=$ sign$($w$,$ x$).$

Note we ignore the y$-$intercept term here, so we only need the optimal w in Rd$.$

$1$

$$ It turns out the correct function to minimize to find the weights is

F $($w$)=1$

N

N$$

i$=1$

log

$($

$1+$ e$$w$,$xi$$yi

$)$

$.$

Questions:

$($a$)$ Is F $($w$)$ a convex function?

$($b$)$ Write down the gradient descent algorithm for minimizing F $.$ You need to

specify the starting point, the formula of the gradient, but you do not need

to specify the exact choice of the stepsize.