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The Risk Quadrangle in Optimization: An Overview with Recent Results and Extensions

Bogdan Grechuk, Anton Malandii, Terry Rockafellar, Stan Uryasev

Abstract

This paper revisits and extends the 2013 development by Rockafellar and Uryasev of the Risk Quadrangle (RQ) as a unified scheme for integrating risk management, optimization, and statistical estimation. The RQ features four stochastics-oriented functionals -- risk, deviation, regret, and error, along with an associated statistic, and articulates their revealing and in some ways surprising interrelationships and dualizations. Additions to the RQ framework that have come to light since 2013 are reviewed in a synthesis focused on both theoretical advancements and practical applications. New quadrangles -- superquantile, superquantile norm, expectile, biased mean, quantile symmetric average union, and $\varphi$-divergence-based quadrangles -- offer novel approaches to risk-sensitive decision-making across various fields such as machine learning, statistics, finance, and PDE-constrained optimization. The theoretical contribution comes in axioms for ``subregularity'' relaxing ``regularity'' of the quadrangle functionals, which is too restrictive for some applications. The main RQ theorems and connections are revisited and rigorously extended to this more ample framework. Examples are provided in portfolio optimization, regression, and classification, demonstrating the advantages and the role played by duality, especially in ties to robust optimization and generalized stochastic divergences.

The Risk Quadrangle in Optimization: An Overview with Recent Results and Extensions

Abstract

This paper revisits and extends the 2013 development by Rockafellar and Uryasev of the Risk Quadrangle (RQ) as a unified scheme for integrating risk management, optimization, and statistical estimation. The RQ features four stochastics-oriented functionals -- risk, deviation, regret, and error, along with an associated statistic, and articulates their revealing and in some ways surprising interrelationships and dualizations. Additions to the RQ framework that have come to light since 2013 are reviewed in a synthesis focused on both theoretical advancements and practical applications. New quadrangles -- superquantile, superquantile norm, expectile, biased mean, quantile symmetric average union, and -divergence-based quadrangles -- offer novel approaches to risk-sensitive decision-making across various fields such as machine learning, statistics, finance, and PDE-constrained optimization. The theoretical contribution comes in axioms for ``subregularity'' relaxing ``regularity'' of the quadrangle functionals, which is too restrictive for some applications. The main RQ theorems and connections are revisited and rigorously extended to this more ample framework. Examples are provided in portfolio optimization, regression, and classification, demonstrating the advantages and the role played by duality, especially in ties to robust optimization and generalized stochastic divergences.

Paper Structure

This paper contains 21 sections, 20 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Background.
  3. Motivation.
  4. Outline.
  5. Examples of Risk Quadrangles
  6. Theoretical Framework
  7. Definitions and Axioms
  8. Fundamental Theorems and Construction of Quadrangles
  9. Primal Representation
  10. Construction of quadrangles.
  11. Dual Representation and Conjugate Functionals
  12. Parent Functionals and Corresponding Positive Homogeneous Fa-milies
  13. Stochastic divergence as a measure of distance
  14. Generalized Regression and Statistical Estimation
  15. Functional Regression
  16. ...and 6 more sections

Key Result

Theorem 13

Let $({\cal R}, {\cal V}, {\cal D}, {\cal E})$ be a subregular risk quadrangle with a statistic ${\cal S}$. Then (a) The relations ${\cal D}(X) = {\cal R}(X)-{\Bbb{E}} X$ and ${\cal R}(X)= {\Bbb{E}} X+{\cal D}(X)$ give a one-to-one correspondence between subregular measures of risk ${\cal R}$ and su for all $X$. (b) The relations ${\cal E}(X) = {\cal V}(X)-{\Bbb{E}} X$ and $\,{\cal V}(X)= {\Bbb{E}

Theorems & Definitions (72)

  • Example 1: Standard Mean-Based Quadrangle, $\lambda > 0$
  • Example 2: Quantile-Based Quadrangle, $\alpha \in (0,1)$
  • Example 3: CVaR-Based Quadrangle, $\alpha \in (0,1)$
  • Example 4: Quantile Symmetric Average Quadrangle, $\alpha \in (0,1)$
  • Example 5: Quantile Symmetric Average Union Quadrangle, $0 \leq \varepsilon < \\\frac{1}{2}({\rm ess\, sup}(X) - {\rm ess\, inf}(X))$
  • Example 6: Expectile-Based Quadrangle (Asymmetric Mean Squared Error (MSE) Version), $q \in (0,1)$
  • Example 7: Expectile-Based Quadrangle (Piecewise Linear Version), $K>0$
  • Example 8: Mean-Based Quadrangle (Piecewise Linear Version)
  • Example 9: Biased Mean-Based Quadrangle, $x \in {\Bbb{R}}$
  • Example 10: $\varphi$--Divergence-Based Quadrangle, $\beta>0$
  • ...and 62 more