Risk Quadrangle and Robust Optimization Based on $\varphi$-Divergence
Keywords: robust optimization, distributionally robust optimization, convex optimization, regression, classification, risk quadrangle, risk measure, $\varphi$-divergence
Abstract: The Fundamental Risk Quadrangle (FRQ) is a unified framework linking risk management, statistical estimation, and optimization. Distributionally robust optimization (DRO) based on $\varphi$-divergence minimizes the maximal expected loss, where the maximum is over a $\varphi$-divergence uncertainty set. This paper introduces the \emph{extended} $\varphi$-divergence and the extended $\varphi$-divergence quadrangle, which integrates DRO into the FRQ framework. We derive the primal and dual representations of the quadrangle elements (risk, deviation, regret, error, and statistic). The dual representation provides an interpretation of classification, portfolio optimization, and regression as robust optimization based on the extended $\varphi$-divergence. The primal representation offers tractable formulations of these robust optimizations as convex optimization. We provide illustrative examples showing that many common problems, such as least-squares regression, quantile regression, support vector machines, and CVaR optimization, fall within this framework. Additionally, we conduct a case study to visualize the optimal solution of the inner maximization in robust optimization.
Primary Area: optimization
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Submission Number: 13507
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