Sparse Semi-supervised Learning Using Conjugate Functions

In this paper, we propose a general framework for sparse semi-supervised learning, which concerns using a small portion of unlabeled data and a few labeled data to represent target functions and thus has the merit of accelerating function evaluations when predicting the output of a new example. This framework makes use of Fenchel-Legendre conjugates to rewrite a convex insensitive loss involving a regularization with unlabeled data, and is applicable to a family of semi-supervised learning methods such as multi-view co-regularized least squares and single-view Laplacian support vector machines (SVMs). As an instantiation of this framework, we propose sparse multi-view SVMs which use a squared ε-insensitive loss. The resultant optimization is an inf-sup problem and the optimal solutions have arguably saddle-point properties. We present a globally optimal iterative algorithm to optimize the problem. We give the margin bound on the generalization error of the sparse multi-view SVMs, and derive the empirical Rademacher complexity for the induced function class. Experiments on artificial and real-world data show their effectiveness. We further give a sequential training approach to show their possibility and potential for uses in large-scale problems and provide encouraging experimental results indicating the efficacy of the margin bound and empirical Rademacher complexity on characterizing the roles of unlabeled data for semi-supervised learnin

PAC-Bayes Analysis of Multi-view Learning

This paper presents eight PAC-Bayes bounds to analyze the generalization performance of multi-view classifiers. These bounds adopt data dependent Gaussian priors which emphasize classifiers with high view agreements. The center of the prior for the first two bounds is the origin, while the center of the prior for the third and fourth bounds is given by a data dependent vector. An important technique to obtain these bounds is two derived logarithmic determinant inequalities whose difference lies in whether the dimensionality of data is involved. The centers of the fifth and sixth bounds are calculated on a separate subset of the training set. The last two bounds use unlabeled data to represent view agreements and are thus applicable to semi-supervised multi-view learning. We evaluate all the presented multi-view PAC-Bayes bounds on benchmark data and compare them with previous single-view PAC-Bayes bounds. The usefulness and performance of the multi-view bounds are discussed.Comment: 35 page

Top Rank Optimization in Linear Time

Bipartite ranking aims to learn a real-valued ranking function that orders positive instances before negative instances. Recent efforts of bipartite ranking are focused on optimizing ranking accuracy at the top of the ranked list. Most existing approaches are either to optimize task specific metrics or to extend the ranking loss by emphasizing more on the error associated with the top ranked instances, leading to a high computational cost that is super-linear in the number of training instances. We propose a highly efficient approach, titled TopPush, for optimizing accuracy at the top that has computational complexity linear in the number of training instances. We present a novel analysis that bounds the generalization error for the top ranked instances for the proposed approach. Empirical study shows that the proposed approach is highly competitive to the state-of-the-art approaches and is 10-100 times faster