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With deep neural networks providing state-of-the-art machine learning modelsfor numerous machine learning tasks, quantifying the robustness of these modelshas become an important area of research. However, most of the researchliterature merely focuses on the \textit{worst-case} setting where the input ofthe neural network is perturbed with noises that are constrained within an$\ell_p$ ball; and several algorithms have been proposed to compute certifiedlower bounds of minimum adversarial distortion based on such worst-caseanalysis. In this paper, we address these limitations and extend the approachto a \textit{probabilistic} setting where the additive noises can follow agiven distributional characterization. We propose a novel probabilisticframework PROVEN to PRObabilistically VErify Neural networks with statisticalguarantees -- i.e., PROVEN certifies the probability that the classifier'stop-1 prediction cannot be altered under any constrained $\ell_p$ normperturbation to a given input. Importantly, we show that it is possible toderive closed-form probabilistic certificates based on current state-of-the-artneural network robustness verification frameworks. Hence, the probabilisticcertificates provided by PROVEN come naturally and with almost no overhead whenobtaining the worst-case certified lower bounds from existing methods such asFast-Lin, CROWN and CNN-Cert. Experiments on small and large MNIST and CIFARneural network models demonstrate our probabilistic approach can achieve up toaround $75\%$ improvement in the robustness certification with at least a$99.99\%$ confidence compared with the worst-case robustness certificatedelivered by CROWN.

PROVEN: Certifying Robustness of Neural Networks with a Probabilistic Approach