Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming (1903.01287v3)

Abstract: Certifying the safety or robustness of neural networks against input uncertainties and adversarial attacks is an emerging challenge in the area of safe machine learning and control. To provide such a guarantee, one must be able to bound the output of neural networks when their input changes within a bounded set. In this paper, we propose a semidefinite programming (SDP) framework to address this problem for feed-forward neural networks with general activation functions and input uncertainty sets. Our main idea is to abstract various properties of activation functions (e.g., monotonicity, bounded slope, bounded values, and repetition across layers) with the formalism of quadratic constraints. We then analyze the safety properties of the abstracted network via the S-procedure and semidefinite programming. Our framework spans the trade-off between conservatism and computational efficiency and applies to problems beyond safety verification. We evaluate the performance of our approach via numerical problem instances of various sizes.

• The paper introduces a novel method that uses quadratic constraints to abstract activation functions for neural network safety verification.
• It formulates the verification challenge as a semidefinite programming feasibility problem, balancing model conservatism with computational efficiency.
• Numerical evaluations demonstrate that the approach reliably provides tighter safety bounds for diverse network configurations in safety-critical applications.

Safety Verification and Robustness Analysis of Neural Networks via Semidefinite Programming

The paper "Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming" tackles the ongoing challenge of ensuring the safety and robustness of neural networks against input uncertainties and adversarial attacks. The nonlinear and large-scale characteristics of neural networks compromise their analysis, often resulting in their deployment as black-box systems without formal guarantees. This characteristic renders them vulnerable to input perturbations, limiting their utility in safety-critical applications such as autonomous driving and collision avoidance.

The authors propose a novel approach that uses semidefinite programming (SDP) to address this verification challenge for neural networks with general activation functions. This method relies on abstracting the properties of activation functions using quadratic constraints. Specifically, it captures features such as monotonicity, bounded slopes, and values using these constraints. The verification problem is then transformed into an SDP feasibility problem using the S-procedure from robust control theory, a pivotal tool for reasoning about multiple quadratic constraints.

Key Contributions

1. Quadratic Constraint Abstraction: The paper introduces an abstraction scheme leveraging quadratic constraints to characterize the nonlinearities and uncertain inputs in neural networks. This abstraction provides a structured way to analyze and verify safety and robustness properties while managing computational complexity.
2. Semidefinite Programming Formulation: The proposed framework embeds the verification problem within an SDP framework, making it possible to efficiently handle the trade-off between conservatism of the model and computational efficiency. This balance is crucial for deploying neural networks in real-time and safety-critical applications.
3. Numerical Evaluation: The approach is numerically validated on problems of diverse sizes, showcasing its capability to deal with neural network configurations of varying complexity. The results demonstrate that their method offers tighter and more reliable bounds than existing approaches such as mixed-integer programming or linear programming relaxations.
4. Extensions to Broader Safety and Robustness Problems: While the paper primarily focuses on safety verification, its framework is extensible to a wide range of problems, including sensitivity analysis, output set estimation, and the analysis of closed-loop stability in control systems.

Implications and Future Directions

By transforming neural network verification problems into semidefinite programs, this research provides a promising direction for the formal analysis of neural systems. Practically, this method suggests that provable guarantees of neural network models could be integrated into the development of real-world applications, enhancing their safety and reliability. Theoretically, the framework enriches the intersection of optimization and neural network theory by offering insights into how these systems can be safely and efficiently managed.

The introduction of quadratic constraints to abstract activation functions bridges a gap between control theory and machine learning, indicating that further interdisciplinary research could yield even more robust safety verification tools. Future research might extend this work to include convolutional networks or recurrent architectures, increasing its applicability in various machine learning paradigms. Moreover, developing scalable SDP solvers could augment the feasibility of deploying such frameworks in large-scale industrial applications.

Overall, the paper provides a significant contribution to the safety analysis of neural networks, aligning with current efforts to build AI systems that are not only intelligent but also reliable and secure.