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Suppose I use a linear Support Vector Machine with slack variables on a dataset that is linearly separable. Could it happen that the Support Vector Machine reports a solution that does not perfectly separate the classes?

As an illustration: Is the situation in the picture possible for a Support Vector Machine with slack variables? Although there is a "better" boundary that allows perfect classification, the Support Vector Machine goes for a sub-optimal solution that misclassifies two samples.

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Having linear separable data means that an optimal solution exists. The support vector machine is an approach to identify the optimal solution with multiple steps through Lagrange multiplication. This method is guaranteed to give you the optimal solution if there is one.

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  • $\begingroup$ Even with slack variables? In that case, one could imagine the algorithm accepting a few misclassifications to increase the distance between the decision boundary and the support vectors. $\endgroup$ Feb 2 '18 at 14:25
  • $\begingroup$ @EliasStrehle sorry for the late response. It might be helpful to know exactly how you introduce the slack variables. But the principles holds: If the datasets are linearly separable the SVM will find the optimal solution. It is only in cases where there is no optimal solution that slack variables can be used to relax constraints and allow for suboptimal solutions instead of empty results. $\endgroup$
    – Gegenwind
    Feb 14 '18 at 11:04

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