Hierarchical End-to-End Taylor Bounds for Complete Neural Network Verification

2026-05-11

AI Summary

Existing neural network reachability analysis methods are overly conservative due to their neglect of higher-order smoothness information. This work proposes HiTaB, a hierarchical end-to-end verification framework that, for the first time in practice, systematically leverages the Hessian matrix and its Lipschitz continuity to construct unified zeroth-, first-, and second-order Taylor bounds. By efficiently propagating curvature-related Lipschitz constants across layers, HiTaB enables branch-and-bound verification under both ℓ₂ and ℓ∞ input perturbations. The resulting reachability bounds are significantly tighter and more informative, markedly improving the accuracy and practicality of safety verification for smooth neural networks.

Abstract

Reachability analysis of neural networks, which seeks to compute or bound the set of outputs attainable over a given input domain, is central to certifying safety and robustness in learning-enabled physical systems. Since exact reachable set computation is generally intractable, existing methods typically rely on tractable overapproximations. Examining the state of the art for smooth, twice-differentiable networks, we observe that existing approaches exploit at most second-order information and do not systematically leverage higher-order information. In this work, we introduce \textsc{HiTaB}, a novel verification framework that exploits second-order smoothness through both the Hessian, $\nabla^2 f$, and its Lipschitz constant, $L_{\nabla^2 f}$. We further develop a unified hierarchy of zeroth-, first-, and second-order bounds, together with precise conditions under which higher-order approximations yield provable improvements. Our main technical contribution is a compositional procedure for efficiently bounding $L_{\nabla^2 f}$ in deep neural networks via layerwise propagation of curvature bounds. We extend the framework to both $\ell_2$- and $\ell_\infty$-constrained input sets and show how it can be integrated into branch-and-bound verification pipelines. To our knowledge, this is the first practical reachability analysis framework for smooth neural networks that systematically exploits Lipschitz continuity of curvature, leading to tighter and more informative safety certificates.

Problem

• reachability analysis
• neural network verification
• Taylor bounds
• Lipschitz continuity
• Hessian

Innovation

• Taylor bounds
• Hessian Lipschitz constant
• reachability analysis
• neural network verification
• curvature propagation