Sparse Semi-Supervised Learning Using Conjugate Functions

Abstract

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 learning.