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Data-Attributed Adaptive Control Barrier Functions: Safety-Certified Training Data Curation via Influence Analysis

Jiachen Li, Shihao Li, Dongmei Chen

Abstract

Learning-based adaptation of Control Barrier Function (CBF) parameters offers a promising path toward safe autonomous navigation that balances conservatism with performance. Yet the accuracy of the underlying safety predictor is ultimately constrained by training data quality, and no prior work has formally characterized how prediction errors propagate through the adaptive pipeline to degrade closed-loop safety guarantees. We introduce Data-Attributed Adaptive CBF (DA-CBF), a framework that integrates TracIn-based data attribution into adaptive CBF learning. Our theoretical contributions are fourfold: (i) corrected two-sided bounds relating the safety-loss surrogate to the CBF constraint margin; (ii) a safety margin preservation theorem showing that prediction error induces quantifiable margin degradation and, via a smooth parameter selector, yields a genuine closed-loop forward invariance guarantee not conditioned on a fixed trajectory; (iii) a CBF-QP constraint perturbation bound that links prediction accuracy directly to recursive feasibility; and (iv) a principled leave-one-out justification for influence-based data curation under explicit smoothness assumptions. On a DynamicUnicycle2D benchmark, DA-CBF reduces prediction RMSE by 35.6\%, expands the certified safe operating set by 39\%, and achieves collision-free navigation in a 16-obstacle environment where the uncurated baseline incurs 3 collisions.

Data-Attributed Adaptive Control Barrier Functions: Safety-Certified Training Data Curation via Influence Analysis

Abstract

Learning-based adaptation of Control Barrier Function (CBF) parameters offers a promising path toward safe autonomous navigation that balances conservatism with performance. Yet the accuracy of the underlying safety predictor is ultimately constrained by training data quality, and no prior work has formally characterized how prediction errors propagate through the adaptive pipeline to degrade closed-loop safety guarantees. We introduce Data-Attributed Adaptive CBF (DA-CBF), a framework that integrates TracIn-based data attribution into adaptive CBF learning. Our theoretical contributions are fourfold: (i) corrected two-sided bounds relating the safety-loss surrogate to the CBF constraint margin; (ii) a safety margin preservation theorem showing that prediction error induces quantifiable margin degradation and, via a smooth parameter selector, yields a genuine closed-loop forward invariance guarantee not conditioned on a fixed trajectory; (iii) a CBF-QP constraint perturbation bound that links prediction accuracy directly to recursive feasibility; and (iv) a principled leave-one-out justification for influence-based data curation under explicit smoothness assumptions. On a DynamicUnicycle2D benchmark, DA-CBF reduces prediction RMSE by 35.6\%, expands the certified safe operating set by 39\%, and achieves collision-free navigation in a 16-obstacle environment where the uncurated baseline incurs 3 collisions.

Paper Structure

This paper contains 13 sections, 8 theorems, 19 equations, 8 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND PROBLEM FORMULATION
  3. System Dynamics and CBF
  4. Adaptive CBF Pipeline
  5. Safety-Loss Surrogate
  6. Formal Problem Statement
  7. SAFETY-CERTIFIED DATA CURATION
  8. From Prediction Error to Safety Degradation
  9. Influence-Based Curation: Motivation and Justification
  10. Data Attribution Step
  11. Preserved Safety Mechanism
  12. EXPERIMENTS
  13. CONCLUSIONS

Key Result

Proposition 1

Under Assumption asm:domain, define the denominator bounds: Then for all $(x, x_{\text{obs}}, \gamma) \in \mathcal{X}_{\text{op}} \times \Gamma$, the safety loss $\Phi$ and CBF margin $\psi$ satisfy: Inverting eq:two_sided yields a usable error bound depending only on the predicted $\hat{\Phi}$ and domain constants: if $|\hat{\Phi} - \Phi| \leq \epsilon_\Phi$, then where $\hat{\psi}_{\max}$ is

Figures (8)

  • Figure 3: Overview of the DA-CBF pipeline. Stage 1: TracIn-based attribution identifies and removes the top 10% harmful training samples. Stage 2: Curated PENN predicts safety loss with 36% lower RMSE, filtered by JRD and CVaR thresholds. Stage 3: Adaptive CBF-QP achieves collision-free navigation where the baseline incurs 3 collisions.
  • Figure 4: Conceptual illustration of certified region expansion. (a) Under the uncurated baseline, large prediction error $\epsilon$ yields a small certified region $\mathcal{C}(\epsilon)$; the trajectory exits and collides. (b) DA-CBF reduces $\epsilon$ to $\epsilon' < \epsilon$, expanding the certified region $\mathcal{C}(\epsilon') \supset \mathcal{C}(\epsilon)$ and enabling collision-free navigation.
  • Figure 5: Training loss (left) and test RMSE (right). DA-PENN (10% removed) achieves RMSE 2.83 vs. baseline 4.39 (36% reduction).
  • Figure 6: Safety metrics in single-obstacle scenarios. Both adaptive methods perform identically, confirming prediction accuracy is not the bottleneck in simple environments.
  • Figure 7: Trajectory comparison in the standard obstacle environment from kim2025adaptive. Fixed Low ($\gamma{=}0.01$) collides twice; the other three methods reach the goal collision-free.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Remark 1
  • Proposition 1: Two-Sided Bounds on Safety-Loss Surrogate
  • proof
  • Definition 1: Safety-Weighted Risk
  • Lemma 1
  • proof
  • Theorem 1: Safety Margin Preservation Under Prediction Error
  • proof
  • Remark 2: Safety Budget
  • Corollary 1: Probabilistic Forward Invariance
  • ...and 9 more