论文标题
支持向量回归：风险四边形框架
Support Vector Regression: Risk Quadrangle Framework
论文作者
论文摘要
本文研究了风险四边形（RQ）理论框架内的支持向量回归（SVR）。每个RQ都包含四个随机功能 - 错误，遗憾，风险和\ emph {deviation}，由所谓的统计量结合在一起。 RQ框架统一了随机优化，风险管理和统计估计。在此框架内，$ \ varepsilon $ -SVR和$ν$ -SVR均显示为\ emph {vapnik error}的最小化和有条件的价值 - 风险（CVAR）标准。 VAPNIK误差和CVAR规范定义了四边形，其统计量等于两个对称分位数的平均值。因此，RQ理论意味着$ \ varepsilon $ -SVR和$ν$ -SVR是两个对称条件分位数的平均值的无偏估计器。此外，在一般的随机环境中证明了$ \ varepsilon $ -SVR和$ν$ -SVR之间的等价性。此外，SVR被配制为偏差最小化问题。 RQ理论的另一个含义是将$ν$ -SVR作为分配鲁棒回归（DRR）问题的提法。最后，得出了RQ框架内SVR的替代双重公式。理论结果通过案例研究得到验证。
This paper investigates Support Vector Regression (SVR) within the framework of the Risk Quadrangle (RQ) theory. Every RQ includes four stochastic functionals -- error, regret, risk, and \emph{deviation}, bound together by a so-called statistic. The RQ framework unifies stochastic optimization, risk management, and statistical estimation. Within this framework, both $\varepsilon$-SVR and $ν$-SVR are shown to reduce to the minimization of the \emph{Vapnik error} and the Conditional Value-at-Risk (CVaR) norm, respectively. The Vapnik error and CVaR norm define quadrangles with a statistic equal to the average of two symmetric quantiles. Therefore, RQ theory implies that $\varepsilon$-SVR and $ν$-SVR are asymptotically unbiased estimators of the average of two symmetric conditional quantiles. Moreover, the equivalence between $\varepsilon$-SVR and $ν$-SVR is demonstrated in a general stochastic setting. Additionally, SVR is formulated as a deviation minimization problem. Another implication of the RQ theory is the formulation of $ν$-SVR as a Distributionally Robust Regression (DRR) problem. Finally, an alternative dual formulation of SVR within the RQ framework is derived. Theoretical results are validated with a case study.
