Support Vector Regression: Risk Quadrangle Framework
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916503307157504 |
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| author | Malandii, Anton Uryasev, Stan |
| author_facet | Malandii, Anton Uryasev, Stan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09178 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Support Vector Regression: Risk Quadrangle Framework Malandii, Anton Uryasev, Stan Machine Learning Statistics Theory 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. |
| title | Support Vector Regression: Risk Quadrangle Framework |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2212.09178 |