Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis

• URL: http://arxiv.org/abs/2406.04476v1
• Date: Thu, 6 Jun 2024 20:02:49 GMT
• Title: Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis
• Authors: Sina Sharifi, Mahyar Fazlyab,
• Abstract summary: We propose a novel reachability analysis method tailored for neural networks with differentiable activations. 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.
• Score: 6.9060054915724
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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