CP-NCBF: A Conformal Prediction-based Approach to Synthesize Verified Neural Control Barrier Functions

TL;DR

CP-NCBF introduces split-conformal prediction into neural Control Barrier Function (NCBF) synthesis to obtain probabilistic safety guarantees with a tunable safety-accuracy trade-off. The method trains an NCBF with a robustness margin, iteratively calibrating via conformal scores to ensure, with high confidence, that safety constraints hold across the state space using finite samples. Empirical results on autonomous driving and aerial/quadruped tasks show CP-NCBF achieving larger safe regions and better scalability than Lipschitz-regularized and grid-based baselines, while maintaining quantified safety guarantees. This approach enables safer, more efficient deployment of neural controllers in high-dimensional, safety-critical systems by reducing conservatism without sacrificing formal probabilistic assurances.

Abstract

Control Barrier Functions (CBFs) are a practical approach for designing safety-critical controllers, but constructing them for arbitrary nonlinear dynamical systems remains a challenge. Recent efforts have explored learning-based methods, such as neural CBFs (NCBFs), to address this issue. However, ensuring the validity of NCBFs is difficult due to potential learning errors. In this letter, we propose a novel framework that leverages split-conformal prediction to generate formally verified neural CBFs with probabilistic guarantees based on a user-defined error rate, referred to as CP-NCBF. Unlike existing methods that impose Lipschitz constraints on neural CBF-leading to scalability limitations and overly conservative safe sets--our approach is sample-efficient, scalable, and results in less restrictive safety regions. We validate our framework through case studies on obstacle avoidance in autonomous driving and geo-fencing of aerial vehicles, demonstrating its ability to generate larger and less conservative safe sets compared to conventional techniques.

Figures (10)

• Figure 1: Overview of the proposed approach: The proposed methodology consists of three main steps. In the first step, the NCBF, denoted by $h_{\theta}$, is trained using an initial robustness parameter $\psi = 0$. In the second step, the conformal score $\hat{q}$ is computed based on the policy induced by the learned NCBF. In the third step, this conformal score is then used to update the robustness margin $\psi$, which is subsequently employed to retrain the NCBF. This iterative process is repeated until the conformal score converges to zero, thereby ensuring the desired probabilistic safety guarantee.
• Figure 2: Visualization of learned NCBFs for autonomous ground vehicle at $\phi=0$: (Left) Unverified NCBF (without robust loss), (Center) CP-NCBF (our method with 99.9% safety guarantee), and (Right) Verified NCBF tayal2024learning. Bold black ovals denote the zero-level sets. Both CP-NCBF and Verified NCBF avoid overlapping the unsafe region, while the Unverified NCBF does not; additionally, CP-NCBF yields a larger safe region than the Verified NCBF
• Figure 3: Visualization of learned NCBFs for the aerial vehicle at $\phi, \dot{x}_1, \dot{x}_2, \dot{\phi} = 0$: (Left) Unverified NCBF (without robust loss), (Center) CP-NCBF (with 99.5% safety guarantee), and (Right) CP-NCBF (with 99.9% safety guarantee). Bold black lines denote the zero-level sets. As the desired safety guarantees become more stringent, the learned NCBF exhibits increased conservatism. This behavior highlights the flexibility of our proposed framework in enabling a systematic trade-off between safety and performance, tailored to the specific requirements of the system.
• Figure 4: This figure presents a comparative analysis of the empirical safety rates achieved by all evaluated methods. The observed safety rates align closely with the corresponding theoretical guarantees, thereby providing empirical validation of the reliability and robustness of the proposed framework. These findings underscore the potential of our approach for deployment in real-world safety-critical applications. Furthermore, we note that VNCBF could not be computed for the geo-fencing and quadruped navigation tasks due to its reliance on grid-based sampling, which is computationally feasible only for systems up to $5$ dimensions. This limitation further illustrates the scalability advantage of our method, which remains tractable and effective even in high-dimensional settings.
• Figure 5: This figure shows the $\alpha$-$\epsilon$ plots for different numbers of verification samples, $N$, and different values of $\beta$.

Theorems & Definitions (7)

• Definition 1: Safety
• theorem 1: Ames_2017
• theorem 2: Safety quantification of Neural CBF
• Remark 1
• proof
• lemma 1: Split Conformal Prediction angelopoulos2022gentleintroductionconformalprediction
• Remark 2