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I have a question regarding the Hinge Loss function used for classifiers and in general the "max-margin" types of classifiers, it is defined as $$max(0,1-t*y)$$ where $t$ is the intended output, either $-1$ or $1$ and $y$ is the classifier output. The point of this formulation is not only to update the weights when there is a misclassification, but also to update the weights when the classification is correct but by a not big enough margin. In this case the margin is $1$, as the $1-t*y$ part means that $y*t$ must not only be of the same sign but also greater than 1 to not cause a weight update. My question is, is it always $1-$ in the Hinge Loss? Can I change the function to be $$max(0,M-t*y)$$ where $M$ is my preferred margin? Why aren't all Hinge Loss functions defined this way? Why is the general or default margin equal to $1$?

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