# Shifting-based Optimizable Linear Relaxations for General Activation Functions

arXiv: https://arxiv.org/abs/2606.20292 · HTML: https://mlsanity.com/paper/2606.20292/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 18 Jun 2026.

## Builds on (most cited first, 26 of 26)

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- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [YOLOv4: Optimal Speed and Accuracy of Object Detection](https://mlsanity.com/paper/2004.10934/) (2020)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors](https://mlsanity.com/paper/2207.02696/) (2022)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [The Fifth International Verification of Neural Networks Competition (VNN-COMP 2024): Summary and Results](https://mlsanity.com/paper/2412.19985/) (2024)
- [Fastened CROWN: Tightened Neural Network Robustness Certificates](https://mlsanity.com/paper/1912.00574/) (2019)
- [The Third International Verification of Neural Networks Competition (VNN-COMP 2022): Summary and Results](https://mlsanity.com/paper/2212.10376/) (2022)
- [The Fourth International Verification of Neural Networks Competition (VNN-COMP 2023): Summary and Results](https://mlsanity.com/paper/2312.16760/) (2023)
- [The 6th International Verification of Neural Networks Competition (VNN-COMP 2025): Summary and Results](https://mlsanity.com/paper/2512.19007/) (2025)
- [Scalable Polyhedral Verification of Recurrent Neural Networks](https://mlsanity.com/paper/2005.13300/) (2020)
- [Learning Specialized Activation Functions for Physics-informed Neural Networks](https://mlsanity.com/paper/2308.04073/) (2023)
- [Automatic Perturbation Analysis for Scalable Certified Robustness and Beyond](https://mlsanity.com/paper/2002.12920/) (2020)
- [LinSyn: Synthesizing Tight Linear Bounds for Arbitrary Neural Network Activation Functions](https://mlsanity.com/paper/2201.13351/) (2022)
- [LiSHT: Non-Parametric Linearly Scaled Hyperbolic Tangent Activation Function for Neural Networks](https://mlsanity.com/paper/1901.05894/) (2019)
- [On Optimizing Back-Substitution Methods for Neural Network Verification](https://mlsanity.com/paper/2208.07669/) (2022)

## Abstract (arXiv)

The use of neural networks (NNs) is rapidly increasing, including in safety- and security-critical domains. To provide formal guarantees about NN behavior, many verification methods rely on optimizable linear relaxations of activation functions. However, existing techniques depend on hand-crafted relaxations for each activation function. Extension to state-of-the-art activation functions therefore requires substantial manual effort. In contrast, our approach SLiR (Shifting-based Linear Relaxations) is broadly applicable, requiring only a Lipschitz constant or a set of critical points. SLiR parameterizes relaxations by their slope and computes the corresponding offset via a shifting procedure that ensures sound upper and lower bounds over the input domain, enabling efficient optimization while maintaining correctness. Our experiments show that SLiR produces tight relaxations across a wide range of practical activation functions and enables verification of up to 7.8x more properties compared to state-of-the-art methods.