Toward Scalable Risk Analysis for Stochastic Systems Using Extreme Value Theory

We aim to analyze the behaviour of a finite-time stochastic system, whose model is not available, in the context of more rare and harmful outcomes. Standard estimators are not effective in making predictions about such outcomes due to their rarity. Instead, we use Extreme Value Theory (EVT), the theory of the long-term behaviour of normalized maxima of random variables. We quantify risk using the upper-semideviation $\rho(Y) = E(\max{Y - \mu,0})$ of an integrable random variable $Y$ with mean $\mu = E(Y)$. $\rho(Y)$ is the risk-aware part of the common mean-upper-semideviation functional $\mu + \lambda \rho(Y)$ with $\lambda \in [0,1]$. To assess more rare and harmful outcomes, we propose an EVT-based estimator for $\rho(Y)$ in a given fraction of the worst cases. We show that our estimator enjoys a closed-form representation in terms of the popular conditional value-at-risk functional. In experiments, we illustrate the extrapolation power of our estimator using a small number of i.i.d. samples ($<$50). Our approach is useful for estimating the risk of finite-time systems when models are inaccessible and data collection is expensive. The numerical complexity does not grow with the size of the state space.