Abstract We introduce the implicitly constrained least squares (ICLS) classifier, a novel semi-supervised version of the least squares classifier. This classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible labelings of the unlabeled data. Unlike other discriminative semi-supervised methods, this approach does not introduce explicit additional assumptions into the objective function, but leverages implicit assumptions already present in the choice of the supervised least squares classifier. This method can be formulated as a quadratic programming problem and its solution can be found using a simple gradient descent procedure. We prove that, in a limited 1-dimensional setting, this approach never leads to performance worse than the supervised classifier. Experimental results show that also in the general multidimensional case performance improvements can be expected, both in terms of the squared loss that is intrinsic to the classifier and in terms of the expected classification error.
Aside from the work by @cite_18 @cite_34 , another attempt to construct a robust semi-supervised version of a supervised classifier has been made in @cite_31 , which introduces the safe semi-supervised support vector machine (S @math VM). This method is an extension of S @math VM @cite_40 which constructs a set of low-density decision boundaries with the help of the additional unlabeled data, and chooses the decision boundary, which, even in the worst-case, gives the highest gain in performance over the supervised solution. If the low-density assumption holds, this procedure provably increases classification accuracy over the supervised solution. The main difference with the method considered in this paper, however, is that we make no such additional assumptions. We show that even without these assumptions, safe improvements are possible for the least squares classifier.
@cite_40 @cite_18 @cite_34 @cite_31
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We introduce a semi-supervised support vector machine (S3VM) method. Given a training set of labeled data and a working set of unlabeled data, S3VM constructs a support vector machine using both the training and working sets. We use S3VM to solve the transduction problem using overall risk minimization (ORM) posed by Vapnik. The transduction problem is to estimate the value of a classification function at the given points in the working set. This contrasts with the standard inductive learning problem of estimating the classification function at all possible values and then using the fixed function to deduce the classes of the working set data. We propose a general S3VM model that minimizes both the misclassification error and the function capacity based on all the available data. We show how the S3VM model for 1-norm linear support vector machines can be converted to a mixed-integer program and then solved exactly using integer programming. Results of S3VM and the standard 1-norm support vector machine approach are compared on ten data sets. Our computational results support the statistical learning theory results showing that incorporating working data improves generalization when insufficient training information is available. In every case, S3VM either improved or showed no significant difference in generalization compared to the traditional approach. A rather simple semi-supervised version of the equally simple nearest mean classifier is presented. However simple, the proposed approach is of practical interest as the nearest mean classifier remains a relevant tool in biomedical applications or other areas dealing with relatively high-dimensional feature spaces or small sample sizes. More importantly, the performance of our semi-supervised nearest mean classifier is typically expected to improve over that of its standard supervised counterpart and typically does not deteriorate with increasing numbers of unlabeled data. This behavior is achieved by constraining the parameters that are estimated to comply with relevant information in the unlabeled data, which leads, in expectation, to a more rapid convergence to the large-sample solution because the variance of the estimate is reduced. In a sense, our proposal demonstrates that it may be possible to properly train a known classification scheme such that it can benefit from unlabeled data, while avoiding the additional assumptions typically made in semi-supervised learning. We cast a semi-supervised nearest mean classifier, previously introduced by the first author, in a more principled log-likelihood formulation that is subject to constraints. This, in turn, leads us to make the important suggestion to not only investigate error rates of semi-supervised learners but also consider the risk they originally aim to optimize. We demonstrate empirically that in terms of classification error, mixed results are obtained when comparing supervised to semi-supervised nearest mean classification, while in terms of log-likelihood on the test set, the semi-supervised method consistently outperforms its supervised counterpart. Comparisons to self-learning, a standard approach in semi-supervised learning, are included to further clarify the way, in which our constrained nearest mean classifier improves over regular, supervised nearest mean classification. It is usually expected that, when labeled data are limited, the learning performance can be improved by exploiting unlabeled data. In many cases, however, the performances of current semi-supervised learning approaches may be even worse than purely using the limited labeled data. It is desired to have safe semi-supervised learning approaches which never degenerate learning performance by using unlabeled data. In this paper, we focus on semi-supervised support vector machines (S3VMs) and propose S4VMs, i.e., safe S3VMs. Unlike S3VMs which typically aim at approaching an optimal low-density separator, S4VMs try to exploit the candidate low-density separators simultaneously to reduce the risk of identifying a poor separator with unlabeled data. We describe two implementations of S4VMs, and our comprehensive experiments show that the overall performance of S4VMs are highly competitive to S3VMs, while in contrast to S3VMs which degenerate performance in many cases, S4VMs are never significantly inferior to inductive SVMs.