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-∝SVM), which extends the proportional SVM (∝SVM) model to its semisupervised version. To the best of our knowledge, SS-∝SVM is the first attempt to cope with the SLLP problem. Two realizations, alter-SS-∝SVM and conv-SS-∝SVM, which are based on alternating optimization and convex relaxation, respectively, are developed to solve the proposed SS-∝SVM model. Moreover, we design a cutting plane (CP) method to optimize conv-SS-∝SVM with a guaranteed convergence rate and present a fast accelerated proximal gradient method to solve the multiple kernel learning subproblem in conv-SS-∝SVM efficiently. Empirical experiments not only justify the superiority of SS-∝SVM over its supervised counterpart in classification accuracy but also demonstrate the high competitive computational efficiency of the CP optimization of conv-SS-∝SVM.

中文翻译：

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通过合并未标记的数据学习标签比例。

带标签比例的学习（LLP）解决了以下问题：训练数据以袋的形式提供，可以访问训练袋的标签比例，而不是各个训练实例的标签。现有的LLP研究假设所有训练包的标签比例均可用。然而，在许多应用中，用标签比例标记所有训练袋是很费时的，这导致了使用标记袋和未标记袋即半监督LLP（SLLP）进行学习的问题。在这项工作中，我们提出了半监督比例支持向量机（SS-∝SVM），它将比例支持向量机（∝SVM）模型扩展到其半监督版本。据我们所知，SS-∝SVM是应对SLLP问题的首次尝试。两个实现，alter-SS-∝SVM和conv-SS-∝SVM，分别基于交替优化和凸松弛来开发它们，以解决所提出的SS-∝SVM模型。此外，我们设计了一种切割平面（CP）方法来优化conv-SS-∝SVM，并确保收敛速度，并提出了一种快速加速的近端梯度法，以有效解决conv-SS-∝SVM中的多核学习子问题。经验实验不仅证明了SS-∝SVM在分类精度上优于其监督下的同类产品，而且还证明了conv-SS-∝SVM的CP优化具有很高的竞争计算效率。我们设计了一种切割平面（CP）方法，以优化conv-SS-∝SVM并确保收敛速度，并提出了一种快速加速的近端梯度法，以有效解决conv-SS-∝SVM中的多核学习子问题。经验实验不仅证明了SS-∝SVM在分类精度上优于其监督下的同类产品，而且还证明了conv-SS-∝SVM的CP优化具有很高的竞争计算效率。我们设计了一种切割平面（CP）方法，以优化conv-SS-∝SVM并确保收敛速度，并提出了一种快速加速的近端梯度法，以有效解决conv-SS-∝SVM中的多核学习子问题。经验实验不仅证明了SS-∝SVM在分类精度上优于其监督下的同类产品，而且还证明了conv-SS-∝SVM的CP优化具有很高的竞争计算效率。