Abstract Interpretation of Fixpoint Iterators with Applications to Neural Networks (2110.08260v2)

Abstract: We present a new abstract interpretation framework for the precise over-approximation of numerical fixpoint iterators. Our key observation is that unlike in standard abstract interpretation (AI), typically used to over-approximate all reachable program states, in this setting, one only needs to abstract the concrete fixpoints, i.e., the final program states. Our framework targets numerical fixpoint iterators with convergence and uniqueness guarantees in the concrete and is based on two major technical contributions: (i) theoretical insights which allow us to compute sound and precise fixpoint abstractions without using joins, and (ii) a new abstract domain, CH-Zonotope, which admits efficient propagation and inclusion checks while retaining high precision. We implement our framework in a tool called CRAFT and evaluate it on a novel fixpoint-based neural network architecture (monDEQ) that is particularly challenging to verify. Our extensive evaluation demonstrates that CRAFT exceeds the state-of-the-art performance in terms of speed (two orders of magnitude), scalability (one order of magnitude), and precision (25% higher certified accuracies).