  Despite great recent advances achieved by deep neural networks (DNNs), they
are often vulnerable to adversarial attacks. Intensive research efforts have
been made to improve the robustness of DNNs; however, most empirical defenses
can be adaptively attacked again, and the theoretically certified robustness is
limited, especially on large-scale datasets. One potential root cause of such
vulnerabilities for DNNs is that although they have demonstrated powerful
expressiveness, they lack the reasoning ability to make robust and reliable
predictions. In this paper, we aim to integrate domain knowledge to enable
robust learning with the reasoning paradigm. In particular, we propose a
certifiably robust learning with reasoning pipeline (CARE), which consists of a
learning component and a reasoning component. Concretely, we use a set of
standard DNNs to serve as the learning component to make semantic predictions,
and we leverage the probabilistic graphical models, such as Markov logic
networks (MLN), to serve as the reasoning component to enable knowledge/logic
reasoning. However, it is known that the exact inference of MLN (reasoning) is
#P-complete, which limits the scalability of the pipeline. To this end, we
propose to approximate the MLN inference via variational inference based on an
efficient expectation maximization algorithm. In particular, we leverage graph
convolutional networks (GCNs) to encode the posterior distribution during
variational inference and update the parameters of GCNs (E-step) and the
weights of knowledge rules in MLN (M-step) iteratively. We conduct extensive
experiments on different datasets and show that CARE achieves significantly
higher certified robustness compared with the state-of-the-art baselines. We
additionally conducted different ablation studies to demonstrate the empirical
robustness of CARE and the effectiveness of different knowledge integration.
