HiQ-Lip: A Quantum-Classical Hierarchical Method for Global Lipschitz Constant Estimation of ReLU Networks

Haoqi He, Yan XIAO

Keywords: Quantum Computing, Lipschitz Constant, Neural Network, Quantum-Classical Hybrid Method, Coherent Ising Machine, QUBO

TL;DR: HiQ-Lip accelerates tight Lipschitz constant estimation for neural networks using small-scale quantum devices, outperforming state-of-the-art methods in speed.

Abstract:

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

Primary Area: learning theory

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Submission Number: 10131