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

Deep neural networks (DNNs) have been shown lack of robustness, as they are vulnerable to small perturbations on the inputs, which has led to safety concerns on applying DNNs to safety-critical domains. Several verification approaches have been developed to automatically prove or disprove safety properties for 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 activation 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.