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

arXiv: https://arxiv.org/abs/1903.01287 · HTML: https://mlsanity.com/paper/1903.01287/

## Key facts

- Cited by 101 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2019.

## Built on by (most cited first, 30 of 101)

- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Adversarial Robustness of Deep Neural Networks: A Survey from a Formal Verification Perspective](https://mlsanity.com/paper/2206.12227/) (2022)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [Robust Explanation Constraints for Neural Networks](https://mlsanity.com/paper/2212.08507/) (2022)
- [Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularization](https://mlsanity.com/paper/2310.00116/) (2023)
- [Robust Implicit Networks via Non-Euclidean Contractions](https://mlsanity.com/paper/2106.03194/) (2021)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks](https://mlsanity.com/paper/2204.00846/) (2022)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [A Unified View of SDP-based Neural Network Verification through Completely Positive Programming](https://mlsanity.com/paper/2203.03034/) (2022)
- [POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/2304.01218/) (2023)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Efficient Reachability Analysis of Closed-Loop Systems with Neural Network Controllers](https://mlsanity.com/paper/2101.01815/) (2021)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Closing the Loop: A Framework for Trustworthy Machine Learning in Power Systems](https://mlsanity.com/paper/2203.07505/) (2022)
- [Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops](https://mlsanity.com/paper/2307.14938/) (2023)
- [Clip-and-Verify: Linear Constraint-Driven Domain Clipping for Accelerating Neural Network Verification](https://mlsanity.com/paper/2512.11087/) (2025)
- [DeepSplit: Scalable Verification of Deep Neural Networks via Operator Splitting](https://mlsanity.com/paper/2106.09117/) (2021)
- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [Synthesis of Stabilizing Recurrent Equilibrium Network Controllers](https://mlsanity.com/paper/2204.00122/) (2022)

## 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.