# Support Vector Regression: Risk Quadrangle Framework

arXiv: https://arxiv.org/abs/2212.09178 · HTML: https://mlsanity.com/paper/2212.09178/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 21 references that are in the map.
- First posted on arXiv 18 Dec 2022.

## Builds on (most cited first, 1 of 1)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)

## Abstract (arXiv)

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 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 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.