mlsanityData to 7 Oct 2026

Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming

arXiv 1903.01287 ↗First posted 4 Mar 2019math.OC cs.LGcited by 100

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Abstract (arXiv)

Certifying the safety or robustness of neural networks against input uncertainties and adversarial attacks is an emerging challenge in the area of safe machine learning and control. To provide such a guarantee, one must be able to bound the output of neural networks when their input changes within a bounded set. In this paper, we propose a semidefinite programming (SDP) framework to address this problem for feed-forward neural networks with general activation functions and input uncertainty sets. Our main idea is to abstract various properties of activation functions (e.g., monotonicity, bounded slope, bounded values, and repetition across layers) with the formalism of quadratic constraints. We then analyze the safety properties of the abstracted network via the S-procedure and semidefinite programming. Our framework spans the trade-off between conservatism and computational efficiency and applies to problems beyond safety verification. We evaluate the performance of our approach via numerical problem instances of various sizes.

Key facts. Cited by 100 papers in the map (data through 7 Oct 2026). First posted on arXiv 4 Mar 2019.

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Citing papers per year: 2020: 1; 2021: 18; 2022: 24; 2023: 17; 2024: 17; 2025: 16; 2026: 7.

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  2. 2023A Unified Algebraic Perspective on Lipschitz Neural Networkscited by 58
  3. 2021Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustnesscited by 43
  4. 2021Reachability Analysis of Neural Feedback Loopscited by 41
  5. 2022Adversarial Robustness of Deep Neural Networks: A Survey from a Formal Verification Perspectivecited by 41
  6. 2023When Deep Learning Meets Polyhedral Theory: A Surveycited by 39
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  9. 2023Physics-Informed Machine Learning for Modeling and Control of Dynamical Systemscited by 28
  10. 2022Robust Explanation Constraints for Neural Networkscited by 23
  11. 2024Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulationcited by 23
  12. 2021Robust Implicit Networks via Non-Euclidean Contractionscited by 19
38 more
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  3. 2022Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networkscited by 15
  4. 2021Neural Network Verification in Controlcited by 14
  5. 2021Reliably-stabilizing piecewise-affine neural network controllerscited by 13
  6. 2022A Unified View of SDP-based Neural Network Verification through Completely Positive Programmingcited by 12
  7. 2023POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systemscited by 12
  8. 2022Neural network training under semidefinite constraintscited by 11
  9. 2021Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllerscited by 10
  10. 2022Stability Verification of Neural Network Controllers using Mixed-Integer Programmingcited by 10
  11. 2023Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assemblycited by 10
  12. 2024ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networkscited by 10
  13. 2022Closing the Loop: A Framework for Trustworthy Machine Learning in Power Systemscited by 9
  14. 2023Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loopscited by 9
  15. 2025Clip-and-Verify: Linear Constraint-Driven Domain Clipping for Accelerating Neural Network Verificationcited by 9
  16. 2021DeepSplit: Scalable Verification of Deep Neural Networks via Operator Splittingcited by 8
  17. 2021Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approachcited by 8
  18. 2022Synthesis of Stabilizing Recurrent Equilibrium Network Controllerscited by 8
  19. 2024Lipschitz constant estimation for general neural network architectures using control toolscited by 8
  20. 2021Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativitycited by 7
  21. 2024Verification of Geometric Robustness of Neural Networks via Piecewise Linear Approximation and Lipschitz Optimisationcited by 7
  22. 2021Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multiplierscited by 6
  23. 2021Verifying Quantized Neural Networks using SMT-Based Model Checkingcited by 6
  24. 2022Neural Lyapunov Differentiable Predictive Controlcited by 6
  25. 2022Risk Verification of Stochastic Systems with Neural Network Controllerscited by 6
  26. 2025Two-Stage Learning of Stabilizing Neural Controllers via Zubov Sampling and Iterative Domain Expansioncited by 6
  27. 2022Probabilistic Verification of ReLU Neural Networks via Characteristic Functionscited by 5
  28. 2023Convolutional Neural Networks as 2-D systemscited by 5
  29. 2023Learning Dissipative Neural Dynamical Systemscited by 5
  30. 2021QNNVerifier: A Tool for Verifying Neural Networks using SMT-Based Model Checkingcited by 4
  31. 2022Chordal Sparsity for SDP-based Neural Network Verificationcited by 4
  32. 2022Global Performance Guarantees for Neural Network Models of AC Power Flowcited by 4
  33. 2023GPU-Accelerated Verification of Machine Learning Models for Power Systemscited by 4
  34. 2024Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guaranteescited by 4
  35. 2024Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networkscited by 4
  36. 2025Logic Gate Neural Networks are Good for Verificationcited by 4
  37. 2021Reduced-Order Neural Network Synthesis with Robustness Guaranteescited by 3
  38. 2022Stability Analysis of Recurrent Neural Networks by IQC with Copositive Mutiplierscited by 3

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