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In article LIBSVM: A Library for Support Vector Machines there is written, than C-SVC uses loss function:

$$ \frac{1}{2}w^Tw+C\sum\limits_{i=1}^l\xi_i$$

OK, I know, what is $w^Tw$.

But what is $\xi_i$? I know, that it is somehow connected with misclassifications, but is it calculated exactly?

P.S. I don't use any non-linear kernels.

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$\xi_i$ are the slack variables. They are typically nonzero when the 2-class data is non-separable. We are trying the minimize the slack as much as possible (by minimizing their sum, since they are non-negative) along with maximizing the margin ($w^Tw$) term.

Exact calculation: Well, if the convex program has been solved to optimality without any optimization error, then yes, they are calculated exactly.

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  • $\begingroup$ I know, that they are typically nonzero when the 2-class data is non-separable. I know, that we are trying to minimize their sum. But I don't know, what is the loss function they are calculated with. Is it a step function or hinge loss or something else? $\endgroup$
    – Felix
    Oct 14 '13 at 20:59
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    $\begingroup$ Yes, it is the hinge loss. The hinge loss has been removed from the objective and made into a bunch of constraints (their number equalt to the number of examples, $l$ in your notation). In particular, $\max[0,1-y_iw^Tx_i]$ is the loss on example $i$. $\endgroup$ Oct 14 '13 at 21:14
  • $\begingroup$ @Theja can you explain how wTw term is same as maximum margin? $\endgroup$ Feb 19 '16 at 6:45

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