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 @math -fold cross-validation and the @math 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 @math -fold cross-validation.
For the separable case and hard-margin SVM without threshold (that is, separating hyperplanes in the form: @math ), @cite_13 proved the radius-margin bound where @math is the diameter of the smallest sphere containing the training set.
@cite_13
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A comprehensive look at learning and generalization theory. The statistical theory of learning and generalization concerns the problem of choosing desired functions on the basis of empirical data. Highly applicable to a variety of computer science and robotics fields, this book offers lucid coverage of the theory as a whole. Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.