Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916278487220224 |
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| author | Sharifi, Sina Fazlyab, Mahyar |
| author_facet | Sharifi, Sina Fazlyab, Mahyar |
| contents | We propose a novel reachability analysis method tailored for neural networks with differentiable activations. Our idea hinges on a sound abstraction of the neural network map based on first-order Taylor expansion and bounding the remainder. To this end, we propose a method to compute analytical bounds on the network's first derivative (gradient) and second derivative (Hessian). A key aspect of our method is loop transformation on the activation functions to exploit their monotonicity effectively. The resulting end-to-end abstraction locally preserves the derivative information, yielding accurate bounds on small input sets. Finally, we employ a branch and bound framework for larger input sets to refine the abstraction recursively. We evaluate our method numerically via different examples and compare the results with relevant state-of-the-art methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis Sharifi, Sina Fazlyab, Mahyar Machine Learning Systems and Control We propose a novel reachability analysis method tailored for neural networks with differentiable activations. Our idea hinges on a sound abstraction of the neural network map based on first-order Taylor expansion and bounding the remainder. To this end, we propose a method to compute analytical bounds on the network's first derivative (gradient) and second derivative (Hessian). A key aspect of our method is loop transformation on the activation functions to exploit their monotonicity effectively. The resulting end-to-end abstraction locally preserves the derivative information, yielding accurate bounds on small input sets. Finally, we employ a branch and bound framework for larger input sets to refine the abstraction recursively. We evaluate our method numerically via different examples and compare the results with relevant state-of-the-art methods. |
| title | Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis |
| topic | Machine Learning Systems and Control |
| url | https://arxiv.org/abs/2406.04476 |