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 $\nu$-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 $\nu$-SVR are asymptotically unbiased estimators of the average of two symmetric conditional quantiles. Moreover, the equivalence between $\varepsilon$-SVR and $\nu$-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 $\nu$-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.

翻译：本文在风险四边形理论框架下研究支持向量回归。每个风险四边形包含四个随机泛函——误差、遗憾、风险和偏差，它们通过所谓的统计量相互关联。风险四边形框架统一了随机优化、风险管理和统计估计。在此框架下，ε-支持向量回归和ν-支持向量回归分别被证明可简化为Vapnik误差和条件风险价值范数的最小化问题。Vapnik误差与条件风险价值范数所定义的四边形具有等于两个对称分位数平均值的统计量。因此，风险四边形理论表明ε-支持向量回归和ν-支持向量回归是两个对称条件分位数平均值的渐近无偏估计量。此外，在一般随机设定下证明了ε-支持向量回归与ν-支持向量回归的等价性。本文还将支持向量回归表述为偏差最小化问题。风险四边形理论的另一推论是将ν-支持向量回归构建为分布鲁棒回归问题。最后，推导出风险四边形框架下支持向量回归的替代对偶形式。通过案例研究验证了理论结果。