Sampled Branch and Bound method for Transductive Support Vector Machines

A Support Vector Machine (SV M) is a powerful classifier method, already used in many problems, which can be viewed as a convex optimization. In recent years, a considerable attention has been given on semi-supervised learning, differing from traditional supervised learning by making use of unlabelled data. In fact, in many applications like text categorization, collecting labelled examples may cost large human efforts, while vast amounts of unlabelled data are often readily available offering some additional information. However, being expensive to pursue all the data labelled, Transductive Support Vector Machines (TSV M) were introduced when only a small fraction of them may be considered available to the learner. One difficulty we come across with TSV M formulation, is that it turns into a non-convex optimization problem. Hence, several techniques have been proposed to solve it. Each technique has its own disadvantages the most important one being local minima sensitivity. As a result, experiments on TSV M sometimes perform worse than SV M . The performance of a classification algorithm can be measured through the error rate on the unlabelled points and evidences show that algorithm’s achievements depend on the selected data sets. Conversely, the global optimal solution may be disentangled bestowing a Branch and Bound (BB) technique able to solve the non convex problem across an implicit enumeration process. Besides, with this method the time complexity increases exponentially in the number of instances. In this paper, an original theoretical representation of a TSV M in terms of LP-type problem and violator space is given. Hence, an appealing and randomized method is presented with some experimental evidences, able to efficiently overcame some limitations of BB and extend its use to solve a TSV M . The method may exploits the sparsity property of a TSV M making possible the scaling of the time complexity, utterly given by the number of support vectors.