The selection of model's parameters plays an important role in the application of support vector classification (SVC). The commonly used method of selecting model's parameters is the k-fold cross validation with grid search (CV). It is extremely time-consuming because it needs to train a large number of SVC models. In this paper, a new method is proposed to train SVC with the selection of model's parameters. Firstly, training SVC with the selection of model's parameters is modeled as a minimax optimization problem (MaxMin-L2-SVC-NCH), in which the minimization problem is an optimization problem of finding the closest points between two normal convex hulls (L2-SVC-NCH) while the maximization problem is an optimization problem of finding the optimal model's parameters. A lower time complexity can be expected in MaxMin-L2-SVC-NCH because CV is abandoned. A gradient-based algorithm is then proposed to solve MaxMin-L2-SVC-NCH, in which L2-SVC-NCH is solved by a projected gradient algorithm (PGA) while the maximization problem is solved by a gradient ascent algorithm with dynamic learning rate. To demonstrate the advantages of the PGA in solving L2-SVC-NCH, we carry out a comparison of the PGA and the famous sequential minimal optimization (SMO) algorithm after a SMO algorithm and some KKT conditions for L2-SVC-NCH are provided. It is revealed that the SMO algorithm is a special case of the PGA. Thus, the PGA can provide more flexibility. The comparative experiments between MaxMin-L2-SVC-NCH and the classical parameter selection models on public datasets show that MaxMin-L2-SVC-NCH greatly reduces the number of models to be trained and the test accuracy is not lost to the classical models. It indicates that MaxMin-L2-SVC-NCH performs better than the other models. We strongly recommend MaxMin-L2-SVC-NCH as a preferred model for SVC task.

翻译：模型参数的选择在支持向量分类（SVC）的应用中起着重要作用。常用的模型参数选择方法是基于网格搜索的k折交叉验证（CV），但该方法需要训练大量SVC模型，因此极其耗时。本文提出了一种结合模型参数选择的SVC训练新方法。首先，将参数选择与SVC训练联合建模为一个极小极大优化问题（MaxMin-L2-SVC-NCH），其中极小化问题为寻找两个标准凸包之间最近点的优化问题（L2-SVC-NCH），而极大化问题则为寻求最优模型参数的优化问题。由于摒弃了CV方法，MaxMin-L2-SVC-NCH有望实现更低的时间复杂度。随后，提出了一种基于梯度的算法来求解MaxMin-L2-SVC-NCH：该算法采用投影梯度法（PGA）求解L2-SVC-NCH，同时利用动态学习率的梯度上升算法处理极大化问题。为验证PGA在求解L2-SVC-NCH中的优势，我们在给出L2-SVC-NCH的SMO算法及相应KKT条件后，将PGA与著名的序列最小优化（SMO）算法进行了对比。结果表明，SMO算法是PGA的一个特例，因此PGA具有更高的灵活性。在公开数据集上，MaxMin-L2-SVC-NCH与经典参数选择模型的对比实验显示：MaxMin-L2-SVC-NCH大幅减少了待训练模型的数量，且测试准确率不逊于经典模型。这表明MaxMin-L2-SVC-NCH性能优于其他模型。我们强烈推荐将MaxMin-L2-SVC-NCH作为SVC任务的首选模型。