模型选择是支持向量学习的关键问题.已有模型选择方法采用嵌套的双层优化框架,内层执行支持向量学习,外层通过最小化泛化误差的估计进行模型选择.该框架过程复杂,计算效率低.简化传统的双层优化框架,提出一个支持向量学习的多参数同时调节方法,在同一优化过程中实现模型选择和学习器训练.首先,将支持向量学习中的参数和超参数合并为一个参数向量,利用序贯无约束极小化技术(sequential unconstrained minimization technique,简称SUMT)分别改写支持向量分类和回归的有约束优化问题,得到多参数同时调节模型的多元无约束形式定义;然后,证明多参数同时调节模型目标函数的局部Lipschitz连续性及水平集有界性.在此基础上,应用变尺度方法(variable metric method,简称VMM)设计并实现了多参数同时调节算法.进一步地,基于多参数同时调节模型的性质,证明了算法收敛性,对比分析了算法复杂性.最后,实验验证同时调节算法的收敛性,并实验对比同时调节算法的有效性.理论证明和实验分析表明,同时调节方法是一种坚实、高效的支持向量模型选择方法.
Model selection is critical to support vector learning. Previous model selection methods mainly adopt a nested two-layer framework, where the inner layer trains the learner and the outer one conducts model selection by minimizing the estimate of the generalization error. Breaking from this framework, this paper proposes an approach of simultaneously tuning multiple parameters of support vector learning, which integrates model selection and learning into one optimization process. It first combines the parameters and hyperparameters involved in support vector learning into one parameter vector. Then, using sequential unconstrained minimization technique (SUMT), it reformulates the constrained optimization problems for support vector classification (SVC) and support vector regression (SVR) as unconstrained optimization problems to give the simultaneous tuning model of SVC and SVR. In addition, it proves the basic properties of the simultaneous tuning model of SVC and SVR, including the local Lipschitz continuity and the boundedness of their level sets. Further, it develops a simultaneous tuning algorithm to iteratively solve simultaneous tuning model. Finally, it proves the convergence of the developed algorithm based on the basic properties of the simultaneous tuning model and provides analysis on complexity of the algorithm as compared with related approaches. The empirical evaluation on benchmark datasets shows that the proposed simultaneous approach has lower running time complexity and exhibits similar predictive performance as existing approaches. Theoretical and experimental results demonstrate that the simultaneous tuning approach is a sound and efficient model selection approach for support vector learning.