This paper investigates Support Vector Regression (SVR) in the context of the fundamental risk quadrangle paradigm. It is shown that both formulations of SVR, $\varepsilon$-SVR and $\nu$-SVR, correspond to the minimization of equivalent regular error measures (Vapnik error and superquantile (CVaR) norm, respectively) with a regularization penalty. These error measures, in turn, give rise to corresponding risk quadrangles. Additionally, the technique used for the construction of quadrangles serves as a powerful tool in proving the equivalence between $\varepsilon$-SVR and $\nu$-SVR. By constructing the fundamental risk quadrangle, which corresponds to SVR, we show that SVR is the asymptotically unbiased estimator of the average of two symmetric conditional quantiles. Additionally, SVR is formulated as a regular deviation minimization problem with a regularization penalty by invoking Error Shaping Decomposition of Regression. Finally, the dual formulation of SVR in the risk quadrangle framework is derived.
翻译:本文从基本风险四角形范式的角度调查支持矢量递减(SVR) 的基本风险四角形模型, 显示SVR、 $\varepsilon$- SVR 和$\nu$- SVR 的两种配方物, 与相应的常规误差措施( Vapnik 错误和超级偏差( CVaR) 规范) 的最小化相符, 并处以正常化处罚。 这些误差措施反过来又产生相应的风险四角形。 此外, 用于构建四角形模型的技术, 成为证明 $\ varepslon$- SVR 和$\ nu$- SVR 之间的等值的有力工具。 通过构建基本风险四角形方形, 与 SVR 相对应, 我们表明 SVR 是两个对等式有条件四角形的平均值的无中间偏向性估计。 此外, SVR 的双重配置框架是SV Quang 。