Nonlinear regularization path for quadratic loss support vector machines

Fully Empirical and Data-Dependent Stability-Based Bounds

The purpose of this paper is to obtain a fully empirical stability-based bound on the generalization ability of a learning procedure, thus, circumventing some limitations of the structural risk minimization framework. We show that assuming a desirable property of a learning algorithm is sufficient to make data-dependency explicit for stability, which, instead, is usually bounded only in an algorithmic-dependent way. In addition, we prove that a well-known and widespread classifier, like the support vector machine (SVM), satisfies this condition. The obtained bound is then exploited for model selection purposes in SVM classification and tested on a series of real-world benchmarking datasets demonstrating, in practice, the effectiveness of our approach.

In-sample and out-of-sample model selection and error estimation for support vector machines

In-sample approaches to model selection and error estimation of support vector machines (SVMs) are not as widespread as out-of-sample methods, where part of the data is removed from the training set for validation and testing purposes, mainly because their practical application is not straightforward and the latter provide, in many cases, satisfactory results. In this paper, we survey some recent and not-so-recent results of the data-dependent structural risk minimization framework and propose a proper reformulation of the SVM learning algorithm, so that the in-sample approach can be effectively applied. The experiments, performed both on simulated and real-world datasets, show that our in-sample approach can be favorably compared to out-of-sample methods, especially in cases where the latter ones provide questionable results. In particular, when the number of samples is small compared to their dimensionality, like in classification of microarray data, our proposal can outperform conventional out-of-sample approaches such as the cross validation, the leave-one-out, or the Bootstrap methods.