Analyzing Deep Neural Networks with Symbolic Propagation: Towards Higher Precision and Faster Verification (1902.09866v2)

Abstract: Deep neural networks (DNNs) have been shown lack of robustness for the vulnerability of their classification to small perturbations on the inputs. This has led to safety concerns of applying DNNs to safety-critical domains. Several verification approaches have been developed to automatically prove or disprove safety properties of DNNs. However, these approaches suffer from either the scalability problem, i.e., only small DNNs can be handled, or the precision problem, i.e., the obtained bounds are loose. This paper improves on a recent proposal of analyzing DNNs through the classic abstract interpretation technique, by a novel symbolic propagation technique. More specifically, the values of neurons are represented symbolically and propagated forwardly from the input layer to the output layer, on top of abstract domains. We show that our approach can achieve significantly higher precision and thus can prove more properties than using only abstract domains. Moreover, we show that the bounds derived from our approach on the hidden neurons, when applied to a state-of-the-art SMT based verification tool, can improve its performance. We implement our approach into a software tool and validate it over a few DNNs trained on benchmark datasets such as MNIST, etc.

• The paper introduces a symbolic propagation technique that tracks neuron activations as linear expressions to impose tighter constraints than traditional methods.
• The method significantly improves the precision of DNN verification, enabling the certification of properties that were previously unprovable.
• Tool implementation on benchmarks like MNIST demonstrates faster SMT-based verification through reduced case splits and improved computation time.

Analyzing Deep Neural Networks with Symbolic Propagation: Towards Higher Precision and Faster Verification

The paper "Analyzing Deep Neural Networks with Symbolic Propagation: Towards Higher Precision and Faster Verification" presents a new approach that enhances traditional verification methods for deep neural networks (DNNs) using symbolic propagation techniques. This approach aims to address the challenges of precision and scalability inherent in existing DNN verification frameworks, particularly those based on abstract interpretation.

Deep neural networks are increasingly deployed in safety-critical applications, yet their vulnerability to small perturbations in input data raises significant concerns about their robustness and verification. Formal verification methods are imperative to certify DNNs against safety properties, but existing methods often falter due to either computational constraints or imprecision. The authors build on the abstract interpretation method—a static analysis framework traditionally used for program verification—to enhance DNN verification using symbolic propagation tactics.

Key Innovations:

1. Symbolic Propagation Technique: The paper introduces symbolic propagation, wherein the activation values of neurons are forwarded symbolically from input to output layers. This method operates atop abstract domains, allowing for more precise tracking of neuron values as linear expressions instead of numerical intervals. The approach is shown to induce tighter constraints on activation values compared to traditional abstract domains.
2. Improved Precision: The symbolic propagation technique reduces the imprecision of traditional abstract interpretation approaches. The tighter constraints obtained through symbolic propagation have far-reaching implications: they permit verification of properties that would otherwise be unprovable using conventional methods alone.
3. Tool Implementation and Validation: The authors implement their symbolic propagation method within a software tool and validate it using benchmark datasets, such as MNIST. The experimental results demonstrate superiority in precision compared to existing methods, highlighting its efficacy.
4. Performance Enhancement of SMT-Based Verification: The paper leverages the bounds derived from symbolic propagation to improve the performance of state-of-the-art SMT-based verification tools. By reducing the number of case splits required in SMT solvers, the authors achieve significant reductions in computation time.

Experimental Insights:

The authors conducted experiments on DNNs configured for the MNIST dataset and the ACAS Xu aircraft collision avoidance system. Results indicated substantial enhancement in verification precision, with a notable number of cases successfully verified that were previously unreachable. Additionally, the symbolic propagation technique drastically reduced the computation time needed for SMT-based verification approaches, suggesting its potential use as a pre-processing step to accelerate overall verification processes.

Implications and Future Directions:

The introduction of symbolic propagation within the abstract interpretation framework represents a pivotal shift towards achieving higher precision and scalability in DNN verification. The paper’s methods can be extended to various abstract domains, potentially accommodating diverse architectures beyond fully connected networks, such as convolutional neural networks. The insights from this research not only underscore the importance of robust DNN verification in high-stakes environments but also propose streamlined methodologies for practical implementation.

Future work might focus on refining symbolic propagation to accommodate other types of neural network components, such as batch normalization and complex activation functions. Moreover, exploring the integration of symbolic propagation with real-time systems and dynamic neural network reconfigurations could further enhance its applicability to autonomous systems and real-world applications.

In summary, this paper sets a foundation for deploying DNN verification with improved precision and speed, leveraging symbolic propagation as a robust mechanism capable of addressing the critical challenges posed by modern artificial intelligence systems in safety-sensitive domains.