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

arXiv: https://arxiv.org/abs/2101.12086 · HTML: https://mlsanity.com/paper/2101.12086/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2021.

## Built on by (most cited first, 14 of 14)

- [Iterative Reachability Estimation for Safe Reinforcement Learning](https://mlsanity.com/paper/2309.13528/) (2023)
- [Uncertainty-aware Latent Safety Filters for Avoiding Out-of-Distribution Failures](https://mlsanity.com/paper/2505.00779/) (2025)
- [On Dynamic Programming Decompositions of Static Risk Measures in Markov Decision Processes](https://mlsanity.com/paper/2304.12477/) (2023)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [RACER: Epistemic Risk-Sensitive RL Enables Fast Driving with Fewer Crashes](https://mlsanity.com/paper/2405.04714/) (2024)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [Distributionally Time-Varying Online Stochastic Optimization under Polyak-Lojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning](https://mlsanity.com/paper/2309.09411/) (2023)
- [Risk-Averse Decision Making Under Uncertainty](https://mlsanity.com/paper/2109.04082/) (2021)
- [Privacy-Preserving Reinforcement Learning Beyond Expectation](https://mlsanity.com/paper/2203.10165/) (2022)
- [Policy Evaluation in Distributional LQR](https://mlsanity.com/paper/2303.13657/) (2023)
- [On the Fundamental Limitations of Dual Static CVaR Decompositions in Markov Decision Processes](https://mlsanity.com/paper/2507.14005/) (2025)
- [Risk-Aware Distributed Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2304.02005/) (2023)
- [Ensemble RL through Classifier Models: Enhancing Risk-Return Trade-offs in Trading Strategies](https://mlsanity.com/paper/2502.17518/) (2025)
- [Actor-Critic Learning for Risk-Constrained Linear Quadratic Regulation](https://mlsanity.com/paper/2510.22267/) (2025)

## Abstract (arXiv)

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.