A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks

TL;DR

This paper introduces a unified convex relaxation framework for neural network robustness verification, revealing an inherent barrier to achieving tight verification bounds with existing methods.

Contribution

It unifies all LP-relaxed verifiers under a general framework, proves strong duality, and demonstrates a fundamental barrier to tight robustness verification for large classes of neural networks.

Findings

Exact solutions do not significantly improve verification gaps.

Large-scale experiments show an inherent barrier to tight verification.

The framework applies to diverse architectures and nonlinearities.

Abstract

Verification of neural networks enables us to gauge their robustness against adversarial attacks. Verification algorithms fall into two categories: exact verifiers that run in exponential time and relaxed verifiers that are efficient but incomplete. In this paper, we unify all existing LP-relaxed verifiers, to the best of our knowledge, under a general convex relaxation framework. This framework works for neural networks with diverse architectures and nonlinearities and covers both primal and dual views of robustness verification. We further prove strong duality between the primal and dual problems under very mild conditions. Next, we perform large-scale experiments, amounting to more than 22 CPU-years, to obtain exact solution to the convex-relaxed problem that is optimal within our framework for ReLU networks. We find the exact solution does not significantly improve upon the gap between PGD and existing relaxed verifiers for various networks trained normally or robustly on MNIST and CIFAR datasets. Our results suggest there is an inherent barrier to tight verification for the large class of methods captured by our framework. We discuss possible causes of this barrier and potential future directions for bypassing it. Our code and trained models are available at http://github.com/Hadisalman/robust-verify-benchmark .