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.

中文翻译：

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通过合并未标记数据来学习标签比例


标签比例学习（LLP）解决了训练数据作为包提供的问题，其中训练包的标签比例而不是单个训练实例的标签是可访问的。现有的 LLP 研究假设所有训练包的标签比例都是可访问的。然而，在许多应用中，用标签比例标记所有训练袋是非常耗时的，这导致了使用标记和未标记的袋进行学习的问题，即半监督LLP（SLLP）。在这项工作中，我们提出了半监督比例支持向量机（SS- ∝\propto SVM），它将比例SVM（∝\propto SVM）模型扩展到其半监督版本。据我们所知，SS- ∝\propto SVM 是解决 SLLP 问题的首次尝试。开发了两种实现，alter-SS- ∝\propto SVM 和 conv-SS- ∝\propto SVM，分别基于交替优化和凸松弛，以求解所提出的 SS- ∝\propto SVM 模型。此外，我们设计了一种切割平面（CP）方法来优化 conv-SS- ∝\propto SVM，并保证收敛速度，并提出了一种快速加速的近端梯度方法来解决 conv-SS- ∝\propto SVM 中的多核学习子问题高效。实证实验不仅证明了 SS- ∝\propto SVM 在分类精度方面优于其监督对应的 SVM，而且还证明了 conv-SS- ∝\propto SVM CP 优化的高竞争计算效率。