Support vector machines (SVMs) are some of the most successful machine learning models for binary classification problems. Their key idea is maximizing the margin from the data to the hyperplane subject to correct classification on training samples. In the SVM training model, hinge loss is sensitive to label noise and unstable for resampling. Moreover, binary loss is the most natural choice for modeling classification errors. Motivated by this, we focus on the kernel SVM with the -norm hinge loss (referred to as -); this is a composite function of the hinge loss and -norm, which has the potential to address the aforementioned challenges. In consideration of the non-convexity and non-smoothness of the -norm hinge loss, we first characterize the limiting subdifferential of the -norm hinge loss and then derive the equivalent relationship between the proximal stationary point, the Karush–Kuhn–Tucker point, and the local optimal solution of -KSVM. Second, we develop an alternating direction method of multipliers for -KSVM and find that any limit point of the sequence generated by the proposed algorithm is a locally optimal solution. Lastly, experiments on synthetic and real datasets demonstrate that - can achieve comparable accuracy compared to the standard kernel SVMs and that the former generally results in fewer support vectors.

中文翻译：

---

具有[公式省略]范数铰链损失的核支持向量机分类器

支持向量机 (SVM) 是解决二元分类问题最成功的机器学习模型之一。他们的关键思想是最大化数据到超平面的余量，以对训练样本进行正确分类。在SVM训练模型中，铰链损失对标签噪声敏感，并且对于重采样不稳定。此外，二元损失是建模分类错误的最自然选择。受此启发，我们重点关注具有-范数铰链损失（简称-）的核SVM；这是铰链损失和范数的复合函数，有可能解决上述挑战。考虑到-范数铰链损失的非凸性和非光滑性，我们首先刻画-范数铰链损失的极限次微分，然后推导近端驻点、Karush-Kuhn-Tucker点之间的等价关系，以及-KSVM的局部最优解。其次，我们为-KSVM开发了乘子的交替方向方法，并发现该算法生成的序列的任何极限点都是局部最优解。最后，对合成数据集和真实数据集的实验表明，与标准核 SVM 相比，可以达到相当的精度，并且前者通常会产生更少的支持向量。