A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks (1902.08722v5)

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 .

• The paper unifies LP-relaxed verification methods into a single convex relaxation framework covering both primal and dual perspectives.
• The study identifies an inherent barrier that limits the tightness of robustness verification despite algorithmic improvements, with results showing minimal gains over PGD benchmarks.
• Extensive empirical evaluation, consuming over 22 CPU-years, underscores the need for new paradigms to overcome the limitations of current convex relaxation techniques.

Analyzing Convex Relaxation Barriers in Robustness Verification of Neural Networks

The paper "A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks" presents an in-depth exploration of the limitations faced by convex relaxation methods in the verification of neural network robustness against adversarial attacks. The authors focus on unifying existing LP-relaxed verifiers within a comprehensive convex relaxation framework, applicable across various network architectures and nonlinearities.

Key Contributions and Findings

The paper offers several major contributions:

1. Unified Framework: The paper consolidates existing LP-relaxed verification techniques for neural networks into a single convex relaxation framework that accommodates both primal and dual perspectives of verification. This encompasses various approaches, including those operating on the primal view by abstract transformers or bounding nonlinearities, and the dual view, which involves relaxing the problem or leveraging dual optimization.
2. Convex Relaxation Barrier: The research identifies an inherent barrier to achieving tight robustness verification within the established convex relaxation techniques. This finding suggests that existing LP-relaxed frameworks are limited by a theoretical constraint, irrespective of further optimizations in algorithm speed or accuracy.
3. Empirical Evaluation: Through extensive experiments amounting to over 22 CPU-years, the paper evaluates the effectiveness of convex-relaxed verification using deep ReLU networks. The authors demonstrate that the exact solution to the convex-relaxed problem adds minimal improvements over existing methods, as evidenced by the maintained gap between Projected Gradient Descent (PGD) and current relaxed verifiers on MNIST and CIFAR-10 datasets.
4. Algorithmic Relationships: The framework elucidates the relationships between different verification methods, enabling a better understanding of their respective strengths and limitations. It specifically showcases the potential equivalence of certain primal and dual strategies when optimal layer-wise relaxations are employed.

Implications and Future Directions

The findings have several implications for both theoretical understanding and practical application of neural network verification:

• Verification Limitations: The identified convex relaxation barrier provides a theoretical upper bound on the efficacy of current LP-relaxed verification methods. Researchers must acknowledge this limit when developing or testing new verification algorithms within this framework.
• Need for New Paradigms: The barrier points toward the necessity of exploring alternative verification paradigms that transcend the conventional layer-wise convex relaxation scope. Potential future directions include integrating hybrid methods, leveraging SDP relaxations, or exploring non-interactive layers for bypassing identified constraints.
• Robust Training Implications: Although the verification limitation exists, robust training methods can still benefit from relaxed verifiers by producing networks better aligned with the layer-wise relaxation framework. This might not fully eliminate the verification gap but can mitigate it under practical settings.

In conclusion, this paper rigorously frames the limitations of current convex relaxation approaches in verifying neural network robustness. By providing a unified theoretical base, it lays the foundation for future explorations beyond the existing frameworks, challenging the research community to innovate advanced verification methodologies. The findings emphasize the need for a paradigm shift towards techniques that can effectively circumnavigate the convex relaxation barrier.