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Hinge loss is usually defined as $$L(y,\hat{y}) = max(0,1-y\hat{y}) $$

What I don't understand is why are we comparing zero with $1-y\hat{y}$ instead of some other constant. Why not make it $2-y\hat{y}$, or $\sqrt2-y\hat{y}$ or just take $y\hat{y}$, to check if the observation would be on the right side of the hyperplane? Is there any reason behind '1' as a constant?

Thanks

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There's no particular reason. It needs a constant different from zero, and 1 fits nicely due to the fact that anything multiplied by 1 is the same thing. You'd get the same result if you replace it with a different number everywhere and adjust the regularization.

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  • $\begingroup$ But how the behavior of the loss would change if constant is zero? $\endgroup$
    – dmonkoff
    Sep 8, 2019 at 18:55
  • $\begingroup$ If you remove that positive constant, the optimal result is for all the coefficients and predictions to be zero. $\endgroup$ Sep 9, 2019 at 6:11
  • $\begingroup$ Okay, that makes sense, thanks $\endgroup$
    – dmonkoff
    Sep 9, 2019 at 13:15

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