CLAD: Constrained Abstract Domain for Neural Network Verification

2026-09-28

AI Summary

This study addresses the over-relaxation problem in neural network verification caused by the inability of existing abstract domains to precisely handle input regions defined by combined convex constraints. To this end, it proposes the Constrained Lagrangian Abstract Domain (CLAD), the first abstract domain framework supporting arbitrary convex constraints. CLAD leverages Lagrange multipliers to transform the bounding of intersections without closed-form solutions into a computable relaxed optimization problem. By employing a projected primal-dual method combined with automatic differentiation, it alternately updates the input space and multipliers to compute tight over-approximation bounds. Evaluated on 1,944 test instances, CLAD verifies 60% more instances under L2-ball constraints and achieves an overall improvement of 22%, significantly enhancing verification precision and reliability for inputs with complex geometries.

Abstract

Neural network verification (NNV) formally verifies that a network satisfies a specified property for all inputs within a defined region. Modern NNV tools employ abstract domains to compute a sound over-approximation of the network's behavior from the given input region, thus the tightness of these abstractions essentially determines efficiency. A long line of increasingly precise domains has been developed, but they all describe the valid input region in the same restrictive way, e.g., an Lp-norm ball. A practical input region is rarely a simple Lp ball, but rather a combination Lp ball with additional constraints. Verifying a network over such a region with existing abstraction produces a loose over-approximation, which results in either failing to verify a property or spurious counterexamples. We introduce Constrained Lagrangian Abstract Domain (CLAD), a new abstract domain that computes a sound over-approximation of neural networks over input regions defined by a combination of convex constraints. CLAD propagates these constraints and tightens bounds over the true feasible region. However, bounding a neuron over the intersection of these constraints has no closed-form solution, so CLAD relaxes each constraint into the objective with a Lagrange multiplier and solves the resulting max-min problem with a projected primal-dual method, alternating a projected gradient step on the input with a multiplier update. CLAD supports any convex constraint with a subgradient, e.g., from automatic differentiation. We evaluate CLAD on 1,944 instances across four convolutional networks with motion-blur structured perturbations with halfspace or L2-ball constraints. On standard unconstrained Linf property, CLAD verifies as many instances as GCPCROWN at a similar runtime. On constrained properties, CLAD verifies 60\% more instances than GCPCROWN on L2-ball properties, and 22% more in total.

Problem

• Neural Network Verification
• Abstract Domain
• Constrained Input Region
• Over-approximation

Innovation

• Neural Network Verification
• Abstract Domain
• Lagrange Multiplier
• Projected Primal-Dual Method
• Convex Constraints