### Rademacher Complexity Bounds for a Penalized Multiclass Semi-Supervised Algorithm

We propose Rademacher complexity bounds for multiclass classifiers trained with a two-step semi-supervised model. In the first step, the algorithm partitions the partially labeled data and then identifies dense clusters containing $\kappa$ predominant classes using the labeled training examples such that the proportion of their non-predominant classes is below a fixed threshold. In the second step, a classifier is trained by minimizing a margin empirical loss over the labeled training set and a penalization term measuring the disability of the learner to predict the $\kappa$ predominant classes of the identified clusters. The resulting data-dependent generalization error bound involves the margin distribution of the classifier, the stability of the clustering technique used in the first step and Rademacher complexity terms corresponding to partially labeled training data. Our theoretical result exhibit convergence rates extending those proposed in the literature for the binary case, and experimental results on different multiclass classification problems show empirical evidence that supports the theory.

### The Pessimistic Limits and Possibilities of Margin-based Losses in Semi-supervised Learning

Consider a classification problem where we have both labeled and unlabeled data available. We show that for linear classifiers defined by convex margin-based surrogate losses that are decreasing, it is impossible to construct \emph{any} semi-supervised approach that is able to guarantee an improvement over the supervised classifier measured by this surrogate loss on the labeled and unlabeled data. For convex margin-based loss functions that also increase, we demonstrate safe improvements \emph{are} possible.

### Implicitly Constrained Semi-Supervised Least Squares Classification

We introduce a novel semi-supervised version of the least squares classifier. This implicitly constrained least squares (ICLS) 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, our 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. We show this approach 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 certain way, our method never leads to performance worse than the supervised classifier. Experimental results corroborate this theoretical result in the multidimensional case on benchmark datasets, also in terms of the error rate.

### Discriminative Semi-Supervised Feature Selection via Manifold Regularization

We consider the problem of semi-supervised feature selection, where we are given a small amount of labeled examples and a large amount of unlabeled examples. Since a small number of labeled samples are usually insufficient for identifying the relevant features, the critical problem arising from semi-supervised feature selection is how to take advantage of the information underneath the unlabeled data. To address this problem, we propose a novel discriminative semi-supervised feature selection method based on the idea of manifold regularization. The proposed method selects features throughmaximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data. We formulate the proposed feature selection method into a convex-concave optimization problem, where the saddle point corresponds to the optimal solution. To find the optimal solution, the level method, a fairly recent optimization method, is employed. We also present a theoretic proof of the convergence rate for the application of the level method to our problem. Empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi-supervised feature selection method.

### The Pessimistic Limits of Margin-based Losses in Semi-supervised Learning

We show that for linear classifiers defined by convex margin-based surrogate losses that are monotonically decreasing, it is impossible to construct any semi-supervised approach that is able to guarantee an improvement over the supervised classifier measured by this surrogate loss. For non-monotonically decreasing loss functions, we demonstrate safe improvements are possible.