$F(w)=\frac{1}{N}{\displaystyle \underset{i=1}{\overset{N}{}}}log(1+e^{-(\text{:}w,x_i\text{:})v_i})$

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.