# Learning Control Barrier Functions from Expert Demonstrations

arXiv: https://arxiv.org/abs/2004.03315 · HTML: https://mlsanity.com/paper/2004.03315/

## Key facts

- Cited by 92 papers in the map (data through 14 Oct 2026).
- Placed in Safe offline RL, one of 13 directions in Offline RL and imitation learning from data.
- First posted on arXiv 7 Apr 2020.

## Placed in

- Offline RL and imitation learning from data / [Safe offline RL](https://mlsanity.com/offlinerl/safe-offline-rl/)

## Built on by (most cited first, 30 of 92)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Safe Offline Reinforcement Learning with Feasibility-Guided Diffusion Model](https://mlsanity.com/paper/2401.10700/) (2024)
- [Generalizing Safety Beyond Collision-Avoidance via Latent-Space Reachability Analysis](https://mlsanity.com/paper/2502.00935/) (2025)
- [Safety Guardrails for LLM-Enabled Robots](https://mlsanity.com/paper/2503.07885/) (2025)
- [Uncertainty-aware Latent Safety Filters for Avoiding Out-of-Distribution Failures](https://mlsanity.com/paper/2505.00779/) (2025)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Lyapunov Density Models: Constraining Distribution Shift in Learning-Based Control](https://mlsanity.com/paper/2206.10524/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Safe Pontryagin Differentiable Programming](https://mlsanity.com/paper/2105.14937/) (2021)
- [Safe Occlusion-aware Autonomous Driving via Game-Theoretic Active Perception](https://mlsanity.com/paper/2105.08169/) (2021)
- [Safety Certification for Stochastic Systems via Neural Barrier Functions](https://mlsanity.com/paper/2206.01463/) (2022)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Steering Dialogue Dynamics for Robustness against Multi-turn Jailbreaking Attacks](https://mlsanity.com/paper/2503.00187/) (2025)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [In-Distribution Barrier Functions: Self-Supervised Policy Filters that Avoid Out-of-Distribution States](https://mlsanity.com/paper/2301.12012/) (2023)
- [Gameplay Filters: Robust Zero-Shot Safety through Adversarial Imagination](https://mlsanity.com/paper/2405.00846/) (2024)
- [Active Uncertainty Reduction for Safe and Efficient Interaction Planning: A Shielding-Aware Dual Control Approach](https://mlsanity.com/paper/2302.00171/) (2023)
- [Safe Reinforcement Learning on the Constraint Manifold: Theory and Applications](https://mlsanity.com/paper/2404.09080/) (2024)
- [Learning Robust Hybrid Control Barrier Functions for Uncertain Systems](https://mlsanity.com/paper/2101.06492/) (2021)
- [SHARP: Shielding-Aware Robust Planning for Safe and Efficient Human-Robot Interaction](https://mlsanity.com/paper/2110.00843/) (2021)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)
- [SafeFlowMatcher: Safe and Fast Planning using Flow Matching with Control Barrier Functions](https://mlsanity.com/paper/2509.24243/) (2025)
- [Joint Differentiable Optimization and Verification for Certified Reinforcement Learning](https://mlsanity.com/paper/2201.12243/) (2022)
- [Online Control Barrier Functions for Decentralized Multi-Agent Navigation](https://mlsanity.com/paper/2303.04313/) (2023)
- [Differentiable Control Barrier Functions for Vision-based End-to-End Autonomous Driving](https://mlsanity.com/paper/2203.02401/) (2022)
- [Temporal Logic Imitation: Learning Plan-Satisficing Motion Policies from Demonstrations](https://mlsanity.com/paper/2206.04632/) (2022)
- [Learning Differentiable Safety-Critical Control using Control Barrier Functions for Generalization to Novel Environments](https://mlsanity.com/paper/2201.01347/) (2022)
- [Sequential Neural Barriers for Scalable Dynamic Obstacle Avoidance](https://mlsanity.com/paper/2307.03015/) (2023)

## Abstract (arXiv)

Inspired by the success of imitation and inverse reinforcement learning in replicating expert behavior through optimal control, we propose a learning based approach to safe controller synthesis based on control barrier functions (CBFs). We consider the setting of a known nonlinear control affine dynamical system and assume that we have access to safe trajectories generated by an expert - a practical example of such a setting would be a kinematic model of a self-driving vehicle with safe trajectories (e.g., trajectories that avoid collisions with obstacles in the environment) generated by a human driver. We then propose and analyze an optimization-based approach to learning a CBF that enjoys provable safety guarantees under suitable Lipschitz smoothness assumptions on the underlying dynamical system. A strength of our approach is that it is agnostic to the parameterization used to represent the CBF, assuming only that the Lipschitz constant of such functions can be efficiently bounded. Furthermore, if the CBF parameterization is convex, then under mild assumptions, so is our learning process. We end with extensive numerical evaluations of our results on both planar and realistic examples, using both random feature and deep neural network parameterizations of the CBF. To the best of our knowledge, these are the first results that learn provably safe control barrier functions from data.