用量子计算加速神经网络光滑性估计，提升速度与精度。
HiQ-Lip: A Hierarchical Quantum-Classical Method for Global Lipschitz Constant Estimation of ReLU Networks
- 分层图压缩与精炼策略适配量子硬件，将常数估计转为二元优化问题。
- 两层网络256个隐单元时，求解速度翻倍，上界更优。
- 适合关注模型鲁棒性与量子机器学习融合的研究者。
估计神经网络的全局Lipschitz常数对理解与提升其鲁棒性和泛化能力至关重要。然而，精确计算属于NP难问题，现有半定规划（SDP）方法存在内存占用高、处理速度慢等挑战。本文提出HiQ-Lip，一种混合量子-经典分层方法，利用量子计算估算全局Lipschitz常数。通过将问题转化为无约束二次二值优化（QUBO），并采用多级图压缩与精炼策略以适应当前量子硬件限制。在全连接网络上的实验表明，HiQ-Lip不仅提供与现有最优方法相当的估计结果，还显著加速计算过程。在包含256个隐藏神经元的两层网络测试中，其求解速度比现有最佳方法LiPopt快一倍，且给出更紧的上界。这些结果表明小型量子设备在推进神经网络鲁棒性评估方面具有巨大潜力。
原文摘要 · Abstract (English)
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 HiQ-Lip, a hybrid quantum-classical hierarchical method that leverages quantum computing to estimate the global Lipschitz constant. We tackle the estimation by converting it into a Quadratic Unconstrained Binary Optimization 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.
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