Achieving the Tightest Relaxation of Sigmoids for Formal Verification
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Abstract (arXiv)
In the field of formal verification, Neural Networks (NNs) are typically reformulated into equivalent mathematical programs which are optimized over. To overcome the inherent non-convexity of these reformulations, convex relaxations of nonlinear activation functions are typically utilized. Common relaxations (i.e., static linear cuts) of "S-shaped" activation functions, however, can be overly loose, slowing down the overall verification process. In this paper, we derive tuneable hyperplanes which upper and lower bound the sigmoid activation function. When tuned in the dual space, these affine bounds smoothly rotate around the nonlinear manifold of the sigmoid activation function. This approach, termed α-sig, allows us to tractably incorporate the tightest possible, element-wise convex relaxation of the sigmoid activation function into a formal verification framework. We embed these relaxations inside of large verification tasks and compare their performance to LiRPA and α-CROWN, a state-of-the-art verification duo.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 17 of its 25 references that are in the map. First posted on arXiv 20 Aug 2024.
Builds on
17 of 25 references.
References · 17
- 2018On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Modelscited by 322
- 2024TrustLLM: Trustworthiness in Large Language Modelscited by 280
- 2021Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verificationcited by 224
- 2020Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifierscited by 183
- 2019A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networkscited by 124
- 2023First Three Years of the International Verification of Neural Networks Competition (VNN-COMP)cited by 58
- 2024Neural Network Verification with Branch-and-Bound for General Nonlinearitiescited by 47
- 2019Fastened CROWN: Tightened Neural Network Robustness Certificatescited by 39
- 2023The Fourth International Verification of Neural Networks Competition (VNN-COMP 2023): Summary and Resultscited by 38
- 2024From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformerscited by 34
- 2021A Review of Formal Methods applied to Machine Learningcited by 33
- 2023Convex Bounds on the Softmax Function with Applications to Robustness Verificationcited by 19
5 more
- 2022Toward Certified Robustness Against Real-World Distribution Shiftscited by 16
- 2021Neural Network Verification in Controlcited by 14
- 2022Provably Tightest Linear Approximation for Robustness Verification of Sigmoid-like Neural Networkscited by 8
- 2022Global Performance Guarantees for Neural Network Models of AC Power Flowcited by 4
- 2023GPU-Accelerated Verification of Machine Learning Models for Power Systemscited by 4
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