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Methods to certify the robustness of neural networks in the presence of inputuncertainty are vital in safety-critical settings. Most certification methodsin the literature are designed for adversarial or worst-case inputs, butresearchers have recently shown a need for methods that consider random inputnoise. In this paper, we examine the setting where inputs are subject to randomnoise coming from an arbitrary probability distribution. We propose arobustness certification method that lower-bounds the probability that networkoutputs are safe. This bound is cast as a chance-constrained optimizationproblem, which is then reformulated using input-output samples to make theoptimization constraints tractable. We develop sufficient conditions for theresulting optimization to be convex, as well as on the number of samples neededto make the robustness bound hold with overwhelming probability. We show for aspecial case that the proposed optimization reduces to an intuitive closed-formsolution. Case studies on synthetic, MNIST, and CIFAR-10 networksexperimentally demonstrate that this method is able to certify robustnessagainst various input noise regimes over larger uncertainty regions than priorstate-of-the-art techniques.

Data-Driven Certification of Neural Networks with Random Input Noise