Chapter 10CVaR Minimizations in Support Vector Machines

Jun-ya Gotoh1 and Akiko Takeda2

1Chuo University, Japan

2The University of Tokyo, Japan

How to measure the riskiness of a random variable has been a major concern of financial risk management. Among the many possible measures, conditional value at risk (CVaR) is viewed as a promising functional for capturing the characteristics of the distribution of a random variable. CVaR has attractive theoretical properties, and its minimization with respect to involved parameters is often tractable. In portfolio selection especially, the minimization of the empirical CVaR is a linear program. On the other hand, machine learning is based on the so-called regularized empirical risk minimization, where a surrogate of the empirical error defined over the in-sample data is minimized under some regularization of the parameters involved. Considering that both theories deal with empirical risk minimization, it is natural to look at their interaction. In fact, a variant of support vector machine (SVM) known as c10-math-0001-SVM implicitly carries out a certain CVaR minimization, though the relation to CVaR is not clarified at the time of the invention of c10-math-0002-SVM.

This chapter overviews the connections between SVMs and CVaR minimization and suggests further interactions ...

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