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Intensive algorithmic efforts have been made to enable the rapid improvementsof certificated robustness for complex ML models recently. However, currentrobustness certification methods are only able to certify under a limitedperturbation radius. Given that existing pure data-driven statisticalapproaches have reached a bottleneck, in this paper, we propose to integratestatistical ML models with knowledge (expressed as logical rules) as areasoning component using Markov logic networks (MLN, so as to further improvethe overall certified robustness. This opens new research questions aboutcertifying the robustness of such a paradigm, especially the reasoningcomponent (e.g., MLN). As the first step towards understanding these questions,we first prove that the computational complexity of certifying the robustnessof MLN is #P-hard. Guided by this hardness result, we then derive the firstcertified robustness bound for MLN by carefully analyzing different modelregimes. Finally, we conduct extensive experiments on five datasets includingboth high-dimensional images and natural language texts, and we show that thecertified robustness with knowledge-based logical reasoning indeedsignificantly outperforms that of the state-of-the-arts.

Improving Certified Robustness via Statistical Learning with Logical Reasoning