# Wasserstein Distributionally Robust Control Barrier Function using Conditional Value-at-Risk with Differentiable Convex Programming

arXiv: https://arxiv.org/abs/2309.08700 · HTML: https://mlsanity.com/paper/2309.08700/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 24 of 24)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Robust Deep Reinforcement Learning against Adversarial Perturbations on State Observations](https://mlsanity.com/paper/2003.08938/) (2020)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Action Robust Reinforcement Learning and Applications in Continuous Control](https://mlsanity.com/paper/1901.09184/) (2019)
- [Robust Reinforcement Learning for Continuous Control with Model Misspecification](https://mlsanity.com/paper/1906.07516/) (2019)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [EGAD! an Evolved Grasping Analysis Dataset for diversity and reproducibility in robotic manipulation](https://mlsanity.com/paper/2003.01314/) (2020)
- [Certified Adversarial Robustness for Deep Reinforcement Learning](https://mlsanity.com/paper/1910.12908/) (2019)
- [On the Robustness of Safe Reinforcement Learning under Observational Perturbations](https://mlsanity.com/paper/2205.14691/) (2022)
- [Learning Convex Optimization Control Policies](https://mlsanity.com/paper/1912.09529/) (2019)
- [Wasserstein Distributionally Robust Stochastic Control: A Data-Driven Approach](https://mlsanity.com/paper/1812.09808/) (2018)
- [Distributionally Robust Policy Learning via Adversarial Environment Generation](https://mlsanity.com/paper/2107.06353/) (2021)
- [ROMAX: Certifiably Robust Deep Multiagent Reinforcement Learning via Convex Relaxation](https://mlsanity.com/paper/2109.06795/) (2021)
- [Distributionally Robust Chance Constrained Data-enabled Predictive Control](https://mlsanity.com/paper/2006.01702/) (2020)
- [Wasserstein Distributionally Robust Motion Control for Collision Avoidance Using Conditional Value-at-Risk](https://mlsanity.com/paper/2001.04727/) (2020)
- [Group Distributionally Robust Reinforcement Learning with Hierarchical Latent Variables](https://mlsanity.com/paper/2210.12262/) (2022)
- [Data-driven distributionally robust MPC for constrained stochastic systems](https://mlsanity.com/paper/2103.03006/) (2021)
- [Safe and Stable Control Synthesis for Uncertain System Models via Distributionally Robust Optimization](https://mlsanity.com/paper/2210.01341/) (2022)

## Built on by (most cited first, 2 of 2)

- [Constrained Meta-Reinforcement Learning for Adaptable Safety Guarantee with Differentiable Convex Programming](https://mlsanity.com/paper/2312.10230/) (2023)
- [Distributionally Safe Reinforcement Learning under Model Uncertainty: A Single-Level Approach by Differentiable Convex Programming](https://mlsanity.com/paper/2310.02459/) (2023)

## Abstract (arXiv)

Control Barrier functions (CBFs) have attracted extensive attention for designing safe controllers for their deployment in real-world safety-critical systems. However, the perception of the surrounding environment is often subject to stochasticity and further distributional shift from the nominal one. In this paper, we present distributional robust CBF (DR-CBF) to achieve resilience under distributional shift while keeping the advantages of CBF, such as computational efficacy and forward invariance. To achieve this goal, we first propose a single-level convex reformulation to estimate the conditional value at risk (CVaR) of the safety constraints under distributional shift measured by a Wasserstein metric, which is by nature tri-level programming. Moreover, to construct a control barrier condition to enforce the forward invariance of the CVaR, the technique of differentiable convex programming is applied to enable differentiation through the optimization layer of CVaR estimation. We also provide an approximate variant of DR-CBF for higher-order systems. Simulation results are presented to validate the chance-constrained safety guarantee under the distributional shift in both first and second-order systems.