Provable Preimage Under-Approximation for Neural Networks (Full Version)

TL;DR

This paper introduces an efficient algorithm for generating symbolic under-approximations of neural network preimages, enabling global property analysis and robustness verification beyond local robustness checks.

Contribution

It presents a novel, scalable method combining linear relaxation and iterative refinement for preimage under-approximation of neural networks, including a complete algorithm for quantitative verification.

Findings

Validated on high-dimensional MNIST classification task

Provides formal guarantees for quantitative verification

Offers useful robustness insights when standard methods fail

Abstract

Neural network verification mainly focuses on local robustness properties, which can be checked by bounding the image (set of outputs) of a given input set. However, often it is important to know whether a given property holds globally for the input domain, and if not then for what proportion of the input the property is true. To analyze such properties requires computing preimage abstractions of neural networks. In this work, we propose an efficient anytime algorithm for generating symbolic under-approximations of the preimage of any polyhedron output set for neural networks. Our algorithm combines a novel technique for cheaply computing polytope preimage under-approximations using linear relaxation, with a carefully-designed refinement procedure that iteratively partitions the input region into subregions using input and ReLU splitting in order to improve the approximation. Empirically, we validate the efficacy of our method across a range of domains, including a high-dimensional MNIST classification task beyond the reach of existing preimage computation methods. Finally, as use cases, we showcase the application to quantitative verification and robustness analysis. We present a sound and complete algorithm for the former, which exploits our disjoint union of polytopes representation to provide formal guarantees. For the latter, we find that our method can provide useful quantitative information even when standard verifiers cannot verify a robustness property.