Estimating the global Lipschitz constant of neural networks is crucial for understanding and improving their robustness and generalization capabilities. However, precise calculations are NP-hard, and current semidefinite programming (SDP) methods face challenges such as high memory usage and slow processing speeds. In this paper, we propose \textbf{HiQ-Lip}, a hybrid quantum-classical hierarchical method that leverages Coherent Ising Machines (CIMs) to estimate the global Lipschitz constant. We tackle the estimation by converting it into a Quadratic Unconstrained Binary Optimization (QUBO) problem and implement a multilevel graph coarsening and refinement strategy to adapt to the constraints of contemporary quantum hardware. Our experimental evaluations on fully connected neural networks demonstrate that HiQ-Lip not only provides estimates comparable to state-of-the-art methods but also significantly accelerates the computation process. In specific tests involving two-layer neural networks with 256 hidden neurons, HiQ-Lip doubles the solving speed and offers more accurate upper bounds than the existing best method, LiPopt. These findings highlight the promising utility of small-scale quantum devices in advancing the estimation of neural network robustness.

翻译：估计神经网络的全局Lipschitz常数对于理解和提升其鲁棒性与泛化能力至关重要。然而，精确计算是NP难问题，而当前的半定规划方法面临内存占用高、处理速度慢等挑战。本文提出\textbf{HiQ-Lip}，一种混合量子-经典分层方法，利用相干伊辛机来估计全局Lipschitz常数。我们将该估计问题转化为二次无约束二进制优化问题，并采用多层图粗化与细化策略以适应现代量子硬件的约束条件。在全连接神经网络上的实验评估表明，HiQ-Lip不仅能提供与最先进方法相当的估计结果，还显著加速了计算过程。在针对具有256个隐藏神经元的双层神经网络的特定测试中，HiQ-Lip的求解速度较现有最佳方法LiPopt提升了一倍，并给出了更精确的上界。这些发现凸显了小型量子设备在推进神经网络鲁棒性估计方面的潜在应用价值。