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Weighted SVM (or fuzzy SVM) is the most widely used SVM variant owning itseffectiveness to the use of instance weights. Proper selection of the instanceweights can lead to increased generalization performance. In this work, weextend the span error bound theory to weighted SVM and we introduce effectivehyperparameter selection methods for the weighted SVM algorithm. Thesignificance of the presented work is that enables the application of spanbound and span-rule with weighted SVM. The span bound is an upper bound of theleave-one-out error that can be calculated using a single trained SVM model.This is important since leave-one-out error is an almost unbiased estimator ofthe test error. Similarly, the span-rule gives the actual value of theleave-one-out error. Thus, one can apply span bound and span-rule ascomputationally lightweight alternatives of leave-one-out procedure forhyperparameter selection. The main theoretical contributions are: (a) we provethe necessary and sufficient condition for the existence of the span of asupport vector in weighted SVM; and (b) we prove the extension of span boundand span-rule to weighted SVM. We experimentally evaluate the span bound andthe span-rule for hyperparameter selection and we compare them with othermethods that are applicable to weighted SVM: the $K$-fold cross-validation andthe ${\xi}-{\alpha}$ bound. Experiments on 14 benchmark data sets and data setswith importance scores for the training instances show that: (a) the conditionfor the existence of span in weighted SVM is satisfied almost always; (b) thespan-rule is the most effective method for weighted SVM hyperparameterselection; (c) the span-rule is the best predictor of the test error in themean square error sense; and (d) the span-rule is efficient and, for certainproblems, it can be calculated faster than $K$-fold cross-validation.

Span error bound for weighted SVM with applications in hyperparameter selection