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Risk-Averse Certification of Bayesian Neural Networks

Xiyue Zhang, Zifan Wang, Yulong Gao, Licio Romao, Alessandro Abate, Marta Kwiatkowska

TL;DR

This work tackles certification of Bayesian neural networks under uncertainty by introducing RAC-BNN, a risk-averse framework that combines sampling-based output-set approximation with template polytopes and CVaR-based performance evaluation. By constructing provably sound convex approximations of the BNN output space and providing probabilistic guarantees on CVaR estimates, RAC-BNN enables flexible risk-level analysis (via the parameter $\alpha$) and improves tightness and efficiency over state-of-the-art methods. The approach is implemented as a prototype tool and validated on regression and classification benchmarks, showing tighter certified bounds, reduced computation time, and robust handling of worst-case scenarios. This risk-aware certification has practical impact for deploying BNNs in safety- and security-critical settings, and the framework can be extended to closed-loop decision systems in future work.

Abstract

In light of the inherently complex and dynamic nature of real-world environments, incorporating risk measures is crucial for the robustness evaluation of deep learning models. In this work, we propose a Risk-Averse Certification framework for Bayesian neural networks called RAC-BNN. Our method leverages sampling and optimisation to compute a sound approximation of the output set of a BNN, represented using a set of template polytopes. To enhance robustness evaluation, we integrate a coherent distortion risk measure--Conditional Value at Risk (CVaR)--into the certification framework, providing probabilistic guarantees based on empirical distributions obtained through sampling. We validate RAC-BNN on a range of regression and classification benchmarks and compare its performance with a state-of-the-art method. The results show that RAC-BNN effectively quantifies robustness under worst-performing risky scenarios, and achieves tighter certified bounds and higher efficiency in complex tasks.

Risk-Averse Certification of Bayesian Neural Networks

TL;DR

This work tackles certification of Bayesian neural networks under uncertainty by introducing RAC-BNN, a risk-averse framework that combines sampling-based output-set approximation with template polytopes and CVaR-based performance evaluation. By constructing provably sound convex approximations of the BNN output space and providing probabilistic guarantees on CVaR estimates, RAC-BNN enables flexible risk-level analysis (via the parameter ) and improves tightness and efficiency over state-of-the-art methods. The approach is implemented as a prototype tool and validated on regression and classification benchmarks, showing tighter certified bounds, reduced computation time, and robust handling of worst-case scenarios. This risk-aware certification has practical impact for deploying BNNs in safety- and security-critical settings, and the framework can be extended to closed-loop decision systems in future work.

Abstract

In light of the inherently complex and dynamic nature of real-world environments, incorporating risk measures is crucial for the robustness evaluation of deep learning models. In this work, we propose a Risk-Averse Certification framework for Bayesian neural networks called RAC-BNN. Our method leverages sampling and optimisation to compute a sound approximation of the output set of a BNN, represented using a set of template polytopes. To enhance robustness evaluation, we integrate a coherent distortion risk measure--Conditional Value at Risk (CVaR)--into the certification framework, providing probabilistic guarantees based on empirical distributions obtained through sampling. We validate RAC-BNN on a range of regression and classification benchmarks and compare its performance with a state-of-the-art method. The results show that RAC-BNN effectively quantifies robustness under worst-performing risky scenarios, and achieves tighter certified bounds and higher efficiency in complex tasks.

Paper Structure

This paper contains 14 sections, 4 theorems, 5 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Formulation
  4. Problem Statement
  5. Methodology
  6. Output Set Approximation
  7. Risk-Averse Evaluation
  8. Experiments
  9. Evaluation Results
  10. RQ1: Is our approach effective in characterising the output set?
  11. RQ2: Is our approach effective in characterising risk-averse performance of BNNs?
  12. RQ3: What is the certification performance of our approach in tightness and efficiency?
  13. RQ4: What is the sample complexity of our approach?
  14. Conclusion

Key Result

proposition 1

The optimal solution $\bm{ \theta}_N^\star$ to the optimisation problem in Equation Opt:quanset is where $[V]_i$ denotes the $i$-th row of $V$. Let $\hat{\mathcal{Y}}_N=\mathcal{H}(\bm{ \theta}_N^\star)$. Given $\epsilon_1\in (0,1)$, $\beta_1\in (0,1)$, and the Euler's constant $\text{e}$, if $N\geq \frac{1}{\epsilon_1} \frac{\text{e}}{\text{e}-1} \Bigl(\ln\frac{1}{\beta_1} + n+L \Bigr)$, then, w

Figures (2)

  • Figure 1: Tail distributions exist in Bayesian neural networks when recognising images with different types of perturbations.
  • Figure 2: Output set computation for Bayesian neural networks when recognising images with different types of perturbations.

Theorems & Definitions (6)

  • definition 1: Bayesian Neural Network
  • proposition 1
  • proof
  • lemma 1
  • lemma 2
  • proposition 2