Improved Concentration Bound for CVaR

Peng Sima, Hao Deng, Xuelin Zhang, Hong Chen

Abstract: Conditional Value at Risk (CVaR) is a generalization of the standard expectation (the optimization objective of traditional machine learning) used to measure the expected loss of extreme events. While previous studies have focused on concentration inequalities for the classical estimator of CVaR with independently and identically distributed (i.i.d.) random variables, many of them are limited to bounded scenarios, provide only unilateral inequalities or attenuate in a polynomial way. To mitigate these limitations, this paper introduces a novel estimator that relies on an estimator of Value at Risk (VaR) and investigates the concentration inequalities in scenarios where the underlying distributions are sub-Gaussian, sub-exponential, or heavy-tailed. Importantly, the inequalities we derive are bilateral, exhibit exponential decay, and are not confined to bounded scenarios. Furthermore, this paper fills the research gap in the CVaR concentration inequalities of the dependent random variables.