A Convex Approach to Validation-Based Learning of the Regularization Constant

Fast-Solving Quasi-Optimal LS-S3VM Based on an Extended Candidate Set

The semisupervised least squares support vector machine (LS-S³VM) is an important enhancement of least squares support vector machines in semisupervised learning. Given that most data collected from the real world are without labels, semisupervised approaches are more applicable than standard supervised approaches. Although a few training methods for LS-S³VM exist, the problem of deriving the optimal decision hyperplane efficiently and effectually has not been solved. In this paper, a fully weighted model of LS-S³VM is proposed, and a simple integer programming (IP) model is introduced through an equivalent transformation to solve the model. Based on the distances between the unlabeled data and the decision hyperplane, a new indicator is designed to represent the possibility that the label of an unlabeled datum should be reversed in each iteration during training. Using the indicator, we construct an extended candidate set consisting of the indices of unlabeled data with high possibilities, which integrates more information from unlabeled data. Our algorithm is degenerated into a special scenario of the previous algorithm when the extended candidate set is reduced into a set with only one element. Two strategies are utilized to determine the descent directions based on the extended candidate set. Furthermore, we developed a novel method for locating a good starting point based on the properties of the equivalent IP model. Combined with the extended candidate set and the carefully computed starting point, a fast algorithm to solve LS-S³VM quasi-optimally is proposed. The choice of quasi-optimal solutions results in low computational cost and avoidance of overfitting. Experiments show that our algorithm equipped with the two designed strategies is more effective than other algorithms in at least one of the following three aspects: 1) computational complexity; 2) generalization ability; and 3) flexibility. However, our algorithm and other algorithms have similar levels of performance in the remaining aspects.

The Q-norm complexity measure and the minimum gradient method: a novel approach to the machine learning structural risk minimization problem

This paper presents a novel approach for dealing with the structural risk minimization (SRM) applied to a general setting of the machine learning problem. The formulation is based on the fundamental concept that supervised learning is a bi-objective optimization problem in which two conflicting objectives should be minimized. The objectives are related to the empirical training error and the machine complexity. In this paper, one general Q-norm method to compute the machine complexity is presented, and, as a particular practical case, the minimum gradient method (MGM) is derived relying on the definition of the fat-shattering dimension. A practical mechanism for parallel layer perceptron (PLP) network training, involving only quasi-convex functions, is generated using the aforementioned definitions. Experimental results on 15 different benchmarks are presented, which show the potential of the proposed ideas.