  Weighted SVM (or fuzzy SVM) is the most widely used SVM variant owning its
effectiveness to the use of instance weights. Proper selection of the instance
weights can lead to increased generalization performance. In this work, we
extend the span error bound theory to weighted SVM and we introduce effective
hyperparameter selection methods for the weighted SVM algorithm. The
significance of the presented work is that enables the application of span
bound and span-rule with weighted SVM. The span bound is an upper bound of the
leave-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 of
the test error. Similarly, the span-rule gives the actual value of the
leave-one-out error. Thus, one can apply span bound and span-rule as
computationally lightweight alternatives of leave-one-out procedure for
hyperparameter selection. The main theoretical contributions are: (a) we prove
the necessary and sufficient condition for the existence of the span of a
support vector in weighted SVM; and (b) we prove the extension of span bound
and span-rule to weighted SVM. We experimentally evaluate the span bound and
the span-rule for hyperparameter selection and we compare them with other
methods that are applicable to weighted SVM: the $K$-fold cross-validation and
the ${\xi}-{\alpha}$ bound. Experiments on 14 benchmark data sets and data sets
with importance scores for the training instances show that: (a) the condition
for the existence of span in weighted SVM is satisfied almost always; (b) the
span-rule is the most effective method for weighted SVM hyperparameter
selection; (c) the span-rule is the best predictor of the test error in the
mean square error sense; and (d) the span-rule is efficient and, for certain
problems, it can be calculated faster than $K$-fold cross-validation.
