Learning With Label Proportions by Incorporating Unmarked Data

Learning with label proportions (LLP) deals with the problem that the training data are provided as bags, where the label proportions of training bags rather than the labels of individual training instances are accessible. Existing LLP studies assume that the label proportions of all training bags are accessible. However, in many applications, it is time-consuming to mark all training bags with label proportions, which leads to the problem of learning with both marked and unmarked bags, namely, semisupervised LLP (SLLP). In this work, we propose semisupervised proportional support vector machine (SS-
$\propto $
SVM), which extends the proportional SVM (
$\propto $
SVM) model to its semisupervised version. To the best of our knowledge, SS-
$\propto $
SVM is the first attempt to cope with the SLLP problem. Two realizations, alter-SS-
$\propto $
SVM and conv-SS-
$\propto $
SVM, which are based on alternating optimization and convex relaxation, respectively, are developed to solve the proposed SS-
$\propto $
SVM model. Moreover, we design a cutting plane (CP) method to optimize conv-SS-
$\propto $
SVM with a guaranteed convergence rate and present a fast accelerated proximal gradient method to solve the multiple kernel learning subproblem in conv-SS-
$\propto $
SVM efficiently. Empirical experiments not only justify the superiority of SS-
$\propto $
SVM over its supervised counterpart in classification accuracy but also demonstrate the high competitive computational efficiency of the CP optimization of conv-SS-
$\propto $
SVM.