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.
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.
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 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.
In this paper, we propose a novel framework for efficiently and accurately estimating Lipschitz constants in hybrid quantum-classical decision models. Our approach integrates classical neural network with quantum variational circuits to address critical issues in learning theory such as fairness verification, robust training, and generalization. By a unified convex optimization formulation, we extend existing classical methods to capture the interplay between classical and quantum layers. This integrated strategy not only provide a tight bound on the Lipschitz constant but also improves computational efficiency with respect to the previous methods.
In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which involves solving a many-variable mixed-integer programs (MIPs), as a practical application. Classically, complete verification techniques struggle with large networks as the combinatorial space grows exponentially, implying that realistic networks are difficult to be verified by classical methods. To alleviate this issue, we propose to use QC for neural network verification and introduce a hybrid quantum procedure to compute provable certificates. By applying Benders decomposition, we split the MIP into a quadratic unconstrained binary optimization and a linear program which are solved by quantum and classical computers, respectively. We further improve existing hybrid methods based on the Benders decomposition by reducing the overall number of iterations and placing a limit on the maximum number of qubits required. We show that, in a simulated environment, our certificate is sound, and provide bounds on the minimum number of qubits necessary to approximate the problem. Finally, we evaluate our method within simulation and on quantum hardware.
In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which involves solving many-variable mixed-integer programs (MIPs), as a practical application. Classically, complete verification techniques struggle with large networks as the combinatorial space grows exponentially, implying that realistic networks are difficult to be verified by classical methods. To alleviate this issue, we propose to use QC for neural network verification and introduce a hybrid quantum procedure to compute provable certificates. By applying Benders decomposition, we split the MIP into a quadratic unconstrained binary optimization and a linear program which are solved by quantum and classical computers, respectively. We further improve existing hybrid methods based on the Benders decomposition by reducing the overall number of iterations and placing a limit on the maximum number of qubits required. We show that, in a simulated environment, our certificate is sound, and provides bounds on the minimum number of qubits necessary to approximate the problem. Finally, we evaluate our method within simulations and on quantum hardware.
Abstract Here in this work, we present a novel Quadratic Binary Optimization (QBO) model for quantized neural network training, enabling the use of arbitrary activation and loss functions through spline interpolation. We introduce Forward Interval Propagation (FIP), a method designed to tackle the challenges of non-linearity and the multi-layer composite structure in neural networks by discretizing activation functions into linear subintervals. This approach preserves the universal approximation properties of neural networks while allowing complex nonlinear functions to be optimized using quantum computers, thus broadening their applicability in artificial intelligence. We provide theoretical upper bounds on the approximation error and the number of Ising spins required, which is independent of the training sample size. A significant challenge in solving the associated Quadratic Constrained Binary Optimization (QCBO) model on a large scale is the presence of numerous constraints. When employing the penalty method to handle these constraints, tuning a large number of penalty coefficients becomes a critical hyperparameter optimization problem, increasing computational complexity and potentially affecting solution quality. To address this, we employ the Quantum Conditional Gradient Descent (QCGD) algorithm, which leverages quantum computing to directly solve the QCBO problem. We prove the convergence of QCGD under a quantum oracle with randomness and bounded variance in objective value, as well as under limited precision constraints in the coefficient matrix. Additionally, we provide an upper bound on the Time-To-Solution for the QCBO solving process. Experimental results using a coherent Ising machine (CIM) demonstrate a 94.95% accuracy on the Fashion MNIST classification task, with comparisons to classical algorithms highlighting superior efficiency. We further validate the robustness of the spline interpolation method and analyze the impact of varying precision levels on QCGD convergence, underscoring the reliability of our framework.
Emerging quantum computing technologies, such as Noisy Intermediate-Scale Quantum (NISQ) devices, offer potential advancements in solving mathematical optimization problems. However, limitations in qubit availability, noise, and errors pose challenges for practical implementation. In this study, we examine two decomposition methods for Mixed-Integer Linear Programming (MILP) designed to reduce the original problem size and utilize available NISQ devices more efficiently. We concentrate on breaking down the original problem into smaller subproblems, which are then solved iteratively using a combined quantum-classical hardware approach. We conduct a detailed analysis for the decomposition of MILP with Benders and Dantzig-Wolfe methods. In our analysis, we show that the number of qubits required to solve Benders is exponentially large in the worst-case, while remains constant for Dantzig-Wolfe. Additionally, we leverage Dantzig-Wolfe decomposition on the use-case of certifying the robustness of ReLU networks. Our experimental results demonstrate that this approach can save up to 90\% of qubits compared to existing methods on quantum annealing and gate-based quantum computers.