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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.
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A Functional Perspective on Learning Symmetric Functions with Neural Networks
Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in many practical applications, including point clouds and particle physics, a relevant notion of generalization should include varying the input size. In this work we treat symmetric functions (of any size) as functions over probability measures, and study the learning and representation of neural networks defined on measures. By focusing on shallow architectures, we establish approximation and generalization bounds under different choices of regularization (such as RKHS and variation norms), that capture a hierarchy of functional spaces with increasing degree of non-linear learning. The resulting models can be learned efficiently and enjoy generalization guarantees that extend across input sizes, as we verify empirically.
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- Award ID(s):
- 1845360
- PAR ID:
- 10233994
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- ISSN:
- 2640-3498
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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