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Neural networks are an increasingly common tool for solving problems that require complex analysis and pattern matching, such as identifying stop signs in a self driving car or processing medical imagery during diagnosis. Accordingly, verification of neural networks for safety and correctness is of great importance, as mispredictions can have catastrophic results in safety critical domains. As neural networks are known to be sensitive to small changes in input, leading to vulnerabilities and adversarial attacks, analyzing the robustness of networks to small changes in input is a key piece of evaluating their safety and correctness. However, there are many real-world scenarios where the requirements of robustness are not clear cut, and it is crucial to develop measures that assess the level of robustness of a given neural network model and compare levels of robustness across different models, rather than using a binary characterization such as robust vs. not robust. We believe there is great need for developing scalable quantitative robustness verification techniques for neural networks. Formal verification techniques can provide guarantees of correctness, but most existing approaches do not provide quantitative robustness measures and are not effective in analyzing real-world network sizes. On the other hand, sampling-based quantitative robustness is not hindered much by the size of networks but cannot provide sound guarantees of quantitative results. We believe more research is needed to address the limitations of both symbolic and sampling-based verification approaches and create sound, scalable techniques for quantitative robustness verification of neural networks. 

Starting with a random initial seed, fuzzers search for inputs that trigger bugs or vulnerabilities. However, fuzzers often fail to generate inputs for program paths guarded by restrictive branch conditions. In this paper, we show that by first identifying rare-paths in programs (i.e., program paths with path constraints that are unlikely to be satisfied by random input generation), and then, generating inputs/seeds that trigger rare-paths, one can improve the coverage of fuzzing tools. In particular, we present techniques 1) that identify rare paths using quantitative symbolic analysis, and 2) generate inputs that can explore these rare paths using path-guided concolic execution. We provide these inputs as initial seed sets to three state of the art fuzzers. Our experimental evaluation on a set of programs shows that the fuzzers achieve better coverage with the rare-path based seed set compared to a random initial seed.