Gradient Descent

$($a$)$ Do all Gradient Descent algorithms lead to the same model provided you let them run long enough?

$($b$)$ Can Gradient Descent get stuck in a local minimum when training a Logistic Regression model?

$($c$)$ Suppose you use Batch Gradient Descent and you plot the validation error at every epoch. If you notice that the validation error consistently goes up$,$ what is likely going on$?$ How can you fix this?

$($d$)$ Is it a good idea to stop Mini$-$batch Gradient Descent immediately when the validation error goes up$?$

$($e$)$ Which Gradient Descent algorithm $($among those we discussed$)$ will reach the vicinity of the optimal solution the fastest? Which will actually converge? How can you make the others converge as well?