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I would like to understand more deeply what the purpose and intuition of slack variables is in support vector machines.

I know that slack variables are used to minimize

$\frac{1}{2} ||w||^2 + C \sum_{n=1}^{N} \xi_n$ and is the distance to the support vectors, but it seems really strange.

I noticed that using a small Penatly constant $C$ allows more error, but why is this? I think it would still be more minimal having less errors as well.

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