auto_LiRPA: An Automatic Linear Relaxation Perturbation Analysis Library for Neural Networks

Title: auto_LiRPA: An Automatic Linear Relaxation Perturbation Analysis Library for Neural Networks Abstract: auto_LiRPA is an automatic linear relaxation perturbation analysis library designed specifically for neural networks and general computation graphs. It can efficiently calculate the upper and lower bounds of neural network outputs, providing a powerful tool for robustness verification and security analysis of neural networks.

Basic Project Information

• Original Author/Maintainer: Verified-Intelligence Team
• Source Platform: GitHub
• Project Link: https://github.com/Verified-Intelligence/auto_LiRPA
• Release Date: 2026-06-10

With the widespread application of deep learning in critical safety domains (such as autonomous driving, medical diagnosis, and financial risk control), robustness verification of neural networks has become a focus of attention in both academia and industry. The "black-box" nature of neural networks makes them vulnerable to adversarial attacks—small input perturbations can lead to completely incorrect model outputs. Therefore, developing tools that can rigorously prove the behavioral boundaries of neural networks within given input perturbation ranges has become crucial. auto_LiRPA is an open-source tool library born in this context. It provides an automated linear relaxation method to analyze the perturbation propagation characteristics of neural networks, laying a mathematical foundation for the formal verification of neural network security.

Linear Relaxation and Perturbation Analysis

The core idea of auto_LiRPA is to convert complex nonlinear neural network operations into manageable linear constraints using linear relaxation technology. For each layer in the neural network, the library calculates the upper and lower bounds of the layer's output relative to input perturbations, which are expressed as linear functions. Specifically, for a perturbation region in the input space (e.g., L-infinity norm ball), auto_LiRPA can:

1. Propagate bounds layer by layer: Starting from the input layer, calculate the upper and lower bounds of each neuron's output layer by layer
2. Handle nonlinear activations: Apply linear relaxation techniques to nonlinear activation functions such as ReLU, Sigmoid, and Tanh
3. Support general computation graphs: Not limited to standard feedforward networks; it also supports complex computation graphs with branches and loops

Automatic Differentiation and Bound Calculation

auto_LiRPA uses automatic differentiation technology to efficiently compute gradient information of boundary functions. This allows it to not only be suitable for forward boundary propagation but also to combine with optimization-based verification methods to tighten boundary estimates through iterative optimization.

Wide Network Architecture Support

auto_LiRPA supports various mainstream neural network architectures:

• Convolutional Neural Networks (CNN): Common architecture for image classification tasks
• Recurrent Neural Networks (RNN/LSTM/GRU): Suitable for sequence data modeling
• Transformer Architecture: Supports computation graph analysis of attention mechanisms
• Residual Networks (ResNet): Handles boundary propagation challenges brought by skip connections

Flexible Relaxation Strategies

The library implements multiple linear relaxation strategies, and users can choose based on verification accuracy and computational efficiency requirements:

• IBP (Interval Bound Propagation): Fast but relatively loose boundary estimation
• CROWN: Tighter boundaries achieved through linear relaxation
• α-CROWN: Introduces learnable parameters to optimize boundary tightness
• Hybrid Strategy: Combines the advantages of multiple methods to balance accuracy and speed

Integration with Deep Learning Frameworks

auto_LiRPA is designed to seamlessly integrate with PyTorch. Users can easily add robustness verification functions to their existing PyTorch code. This integration method lowers the barrier to use, allowing researchers and engineers to quickly introduce formal verification into practical projects.

Adversarial Robustness Verification

In the field of adversarial machine learning, auto_LiRPA can be used to verify whether a model maintains correct classification within a specific perturbation range. For example, for an image classification model, it can be proven that the model's output label will not change when pixel values change by no more than ε. This formal guarantee is more reliable than traditional adversarial testing methods.

Safety Verification of Neural Network Controllers

In reinforcement learning and control applications, neural networks are often used as policy networks or value functions. auto_LiRPA can help verify the behavioral boundaries of these controllers under state space perturbations, ensuring the safety of control systems in uncertain environments.

Model Interpretation and Sensitivity Analysis

By calculating the boundary sensitivity of outputs to each dimension of inputs, auto_LiRPA can also be used for model interpretation. It can identify the input features that have the greatest impact on prediction results, helping to understand the model's decision-making basis.

Although auto_LiRPA has made significant progress in the field of neural network verification, it still faces some challenges:

1. Scalability: For ultra-large-scale networks (such as large vision models and large language models), the computational cost of boundary calculation is still high
2. Tightness: Boundary estimates for some complex network structures may be too loose, affecting the practicality of verification
3. Dynamic Networks: Support for networks with dynamic control flows (such as conditional execution and data-dependent loops) still needs improvement

Future development directions may include:

• Using GPU parallel computing to accelerate boundary propagation
• Combining abstract interpretation technology to improve boundary tightness
• Extending to probabilistic neural networks and Bayesian inference scenarios

As an open-source neural network perturbation analysis library, auto_LiRPA provides important infrastructure for the formal verification of deep learning systems. It transforms complex mathematical theories into practical engineering tools, enabling researchers and engineers to apply neural network robustness verification techniques in practical projects. With the in-depth application of deep learning in safety-critical domains, the importance of such tools will continue to grow.