Question 30 logistic regression

Can someone explain why in question 30 b) we get to logistic regression ? I'm struggling a bit ... To me it would seem that having the sign function as activation function for the output layer makes y be -1 or 1 (or 0) but not {0, 1} as we should have in \(y = argmax_{y\in \{0, 1\}} p(Y = y | x)\). As I analyzed the inner of the sign function, to me the problem was to study the sign of \(\alpha_1 x1 + \alpha_2 x2\), so not exactly what was expected...
Thanks !

iirc you take the sign because you want to map the probability output from the sigmoid [0,1] - 1/2 to the class labels that are presumably {-1,1} in this case

I feel like this correct answer is, that without -1/2 and the sign function, it should have been logistic regression. With it, its basically logistic "classification".

I am also confused..

Is it because that
when 1/(1+exp(-a)) > 1/2 , sign function outputs 1.
when 1/(1+exp(-a)) < 1/2, sign function outputs -1.
So we consider it as a binary classification, but the y lable is {1,-1}, I don't know why the answer is logistic regression with lable {0,1}.

Can someone help me?
Thanks in advance!

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