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Say I have a population with classes c1 and c2 where size(c1) >> size(c2)

Does the SVM suffer from imbalanced classes sizes where I need to manually correct the training procedure?

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By default, SVM is susceptible to class imbalance. In many implementations, the misclassification penalties can indeed be reweighted to account for an unequal number of samples, but this usually has to be 'turned on'. See Does Support Vector Machine handle imbalanced Dataset?, SVM for unbalanced data

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No. The imbalanced classes are weighted inside the algorithim in such a way that if you had a label that was only 2% observations that it would be accounted for. This is why it's a common used learner for rare events.

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