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Counterexample Guided Branching via Directional Relaxation Analysis in Complete Neural Network Verification

Jingyang Li, Fu Song, Guoqiang Li

Abstract

Deep Neural Networks demonstrate exceptional performance but remain vulnerable to adversarial perturbations, necessitating formal verification for safety-critical deployment. To address the computational complexity of this task, researchers often employ abstraction-refinement techniques that iteratively tighten an over-approximated model. While structural methods utilize Counterexample-Guided Abstraction Refine- ment, state-of-the-art dataflow verifiers typically rely on Branch-and-Bound to refine numerical convex relaxations. However, current dataflow approaches operate with blind refinement processes that rely on static heuristics and fail to leverage specific diagnostic information from verification failures. In this work, we argue that Branch-and-Bound should be reformulated as a Dataflow CEGAR loop where the spurious counterexample serves as a precise witness to local abstraction errors. We propose DRG-BaB, a framework that introduces the Directional Relaxation Gap heuristic to prioritize branching on neurons actively contributing to falsification in the abstract domain. By deriving a closed-form spurious counterexample directly from linear bounds, our method transforms generic search into targeted refinement. Experiments on high-dimensional benchmarks demonstrate that this approach significantly reduces search tree size and verification time compared to established baselines.

Counterexample Guided Branching via Directional Relaxation Analysis in Complete Neural Network Verification

Abstract

Deep Neural Networks demonstrate exceptional performance but remain vulnerable to adversarial perturbations, necessitating formal verification for safety-critical deployment. To address the computational complexity of this task, researchers often employ abstraction-refinement techniques that iteratively tighten an over-approximated model. While structural methods utilize Counterexample-Guided Abstraction Refine- ment, state-of-the-art dataflow verifiers typically rely on Branch-and-Bound to refine numerical convex relaxations. However, current dataflow approaches operate with blind refinement processes that rely on static heuristics and fail to leverage specific diagnostic information from verification failures. In this work, we argue that Branch-and-Bound should be reformulated as a Dataflow CEGAR loop where the spurious counterexample serves as a precise witness to local abstraction errors. We propose DRG-BaB, a framework that introduces the Directional Relaxation Gap heuristic to prioritize branching on neurons actively contributing to falsification in the abstract domain. By deriving a closed-form spurious counterexample directly from linear bounds, our method transforms generic search into targeted refinement. Experiments on high-dimensional benchmarks demonstrate that this approach significantly reduces search tree size and verification time compared to established baselines.
Paper Structure (30 sections, 6 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 30 sections, 6 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neural Networks and Verification Problem
  4. Linear Relaxation and Branch-and-Bound Verification
  5. Motivation
  6. The Limitation of Global and Average-case Information.
  7. A Case Study: The "Wide but Tight" Trap.
  8. Branching as Counter-Example Guided Refinement.
  9. Methodology
  10. Framework Overview
  11. Identifying the Witness: Spurious Counterexample Generation
  12. Option 1: Exact Linear Programming (LP).
  13. Option 2: Adversarial Attacks.
  14. Option 3: Closed-Form Construction from Computed Bounds (Ours).
  15. Verdict.
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Visualizing the Intuition of Solver-Aware Branching. (a) When $\mathbf{A} \ge 0$, the lower bound is active. In $\alpha$-CROWN, this bound is defined by an adjustable slope $\alpha$, allowing the solver to dynamically reduce the error. (b) When $\mathbf{A} < 0$, the upper bound is active. This bound is the McCormick triangle, which is structurally static. Branching is the only mechanism to tighten this rigid geometry.
  • Figure 2: Verification of the Orthogonality Hypothesis (RQ3). The cactus plots show the cumulative number of solved instances versus the number of visited branches (log scale) for different CIFAR-10 architectures.