Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis
<|reference_start|>Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis: 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.<|reference_end|>
@article{sharifi2024provable,
    title={Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving
  Reachability Analysis},
    author={Sina Sharifi, Mahyar Fazlyab},
    journal={arXiv preprint arXiv:2406.04476},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.04476},
    primaryClass={cs.LG cs.SY eess.SY}
}