Risk-sensitive safety analysis using Conditional Value-at-Risk

TL;DR

This paper introduces a new method for safety analysis of stochastic systems that accounts for rare, severe outcomes by using Conditional Value-at-Risk, providing computationally feasible safe set approximations.

Contribution

It develops tractable under-approximations for risk-sensitive safe sets using CVaR, avoiding state space augmentation and requiring only one MDP solution per parameter.

Findings

Provides a guaranteed upper bound on CVaR of maximum cost.

Introduces a second, tractable definition of risk-sensitive safe sets.

Demonstrates effectiveness through numerical examples.

Abstract

This paper develops a safety analysis method for stochastic systems that is sensitive to the possibility and severity of rare harmful outcomes. We define risk-sensitive safe sets as sub-level sets of the solution to a non-standard optimal control problem, where a random maximum cost is assessed via Conditional Value-at-Risk (CVaR). The objective function represents the maximum extent of constraint violation of the state trajectory, averaged over a given percentage of worst cases. This problem is well-motivated but difficult to solve tractably because the temporal decomposition for CVaR is history-dependent. Our primary theoretical contribution is to derive computationally tractable under-approximations to risk-sensitive safe sets. Our method provides a novel, theoretically guaranteed, parameter-dependent upper bound to the CVaR of a maximum cost without the need to augment the state space. For a fixed parameter value, the solution to only one Markov decision process problem is required to obtain the under-approximations for any family of risk-sensitivity levels. In addition, we propose a second definition for risk-sensitive safe sets and provide a tractable method for their estimation without using a parameter-dependent upper bound. The second definition is expressed in terms of a new coherent risk functional, which is inspired by CVaR. We demonstrate our primary theoretical contribution via numerical examples.