A Tale of Two Approximations: Tightening Over-Approximation for DNN Robustness Verification via Under-Approximation
This work addresses the reliability and security of deep neural networks in critical systems by providing more accurate formal verification, though it is incremental as it builds on existing approximation methods.
The paper tackles the problem of improving the scalability and accuracy of formal verification for deep neural network robustness by identifying that existing tight approximations rely on approximation domains rather than actual domains, leading to overestimation. The proposed dual-approximation approach, implemented in DualApp, significantly outperforms state-of-the-art methods with up to 1000% improvement in verified robustness ratio and an average 10.64% increase in certified lower bound.
The robustness of deep neural networks (DNNs) is crucial to the hosting system's reliability and security. Formal verification has been demonstrated to be effective in providing provable robustness guarantees. To improve its scalability, over-approximating the non-linear activation functions in DNNs by linear constraints has been widely adopted, which transforms the verification problem into an efficiently solvable linear programming problem. Many efforts have been dedicated to defining the so-called tightest approximations to reduce overestimation imposed by over-approximation. In this paper, we study existing approaches and identify a dominant factor in defining tight approximation, namely the approximation domain of the activation function. We find out that tight approximations defined on approximation domains may not be as tight as the ones on their actual domains, yet existing approaches all rely only on approximation domains. Based on this observation, we propose a novel dual-approximation approach to tighten over-approximations, leveraging an activation function's underestimated domain to define tight approximation bounds. We implement our approach with two complementary algorithms based respectively on Monte Carlo simulation and gradient descent into a tool called DualApp. We assess it on a comprehensive benchmark of DNNs with different architectures. Our experimental results show that DualApp significantly outperforms the state-of-the-art approaches with 100% - 1000% improvement on the verified robustness ratio and 10.64% on average (up to 66.53%) on the certified lower bound.