Thus our heuristic relaxation does give valid upper bounds of the exact Lipschitz constant.

Fast and precise Lipschitz constant estimation of neural networks is an important task for deep learning. Researchers have recently found an intrinsic trade-off between the accuracy and smoothness of neural networks, so training a network with a loose Lipschitz constant estimation imposes a strong regularization, and can hurt the model accuracy significantly. In this work, we provide a unified theoretical framework, a quantitative geometric approach, to address the Lipschitz constant estimation. By adopting this framework, we can immediately obtain several theoretical results, including the computational hardness of Lipschitz constant estimation and its approximability. We implement the algorithms induced from this quantitative geometric approach, which are based on semidefinite programming (SDP). Our empirical evaluation demonstrates that they are more scalable and precise than existing tools on Lipschitz constant estimation for $\ell_\infty$-perturbations. Furthermore, we also show their intricate relations with other recent SDP-based techniques, both theoretically and empirically. We believe that this unified quantitative geometric perspective can bring new insights and theoretical tools to the investigation of neural-network smoothness and 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 \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.

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.

Fast and precise Lipschitz constant estimation of neural networks is an important task for deep learning. Researchers have recently found an intrinsic trade-off between the accuracy and smoothness of neural networks, so training a network with a loose Lipschitz constant estimation imposes a strong regularization, and can hurt the model accuracy significantly. In this work, we provide a unified theoretical framework, a quantitative geometric approach, to address the Lipschitz constant estimation. By adopting this framework, we can immediately obtain several theoretical results, including the computational hardness of Lipschitz constant estimation and its approximability. We implement the algorithms induced from this quantitative geometric approach, which are based on semidefinite programming (SDP).

We propose a novel layer-wise parameterization for convolutional neural networks (CNNs) that includes built-in robustness guarantees by enforcing a prescribed Lipschitz bound. Each layer in our parameterization is designed to satisfy a linear matrix inequality (LMI), which in turn implies dissipativity with respect to a specific supply rate. Collectively, these layer-wise LMIs ensure Lipschitz boundedness for the input-output mapping of the neural network, yielding a more expressive parameterization than through spectral bounds or orthogonal layers. Our new method LipKernel directly parameterizes dissipative convolution kernels using a 2-D Roesser-type state space model. This means that the convolutional layers are given in standard form after training and can be evaluated without computational overhead. In numerical experiments, we show that the run-time using our method is orders of magnitude faster than state-of-the-art Lipschitz-bounded networks that parameterize convolutions in the Fourier domain, making our approach particularly attractive for improving robustness of learning-based real-time perception or control in robotics, autonomous vehicles, or automation systems. We focus on CNNs, and in contrast to previous works, our approach accommodates a wide variety of layers typically used in CNNs, including 1-D and 2-D convolutional layers, maximum and average pooling layers, as well as strided and dilated convolutions and zero padding. However, our approach naturally extends beyond CNNs as we can incorporate any layer that is incrementally dissipative.

This paper is devoted to the estimation of the Lipschitz constant of neural networks using semidefinite programming. For this purpose, we interpret neural networks as time-varying dynamical systems, where the $k$-th layer corresponds to the dynamics at time $k$. A key novelty with respect to prior work is that we use this interpretation to exploit the series interconnection structure of neural networks with a dynamic programming recursion. Nonlinearities, such as activation functions and nonlinear pooling layers, are handled with integral quadratic constraints. If the neural network contains signal processing layers (convolutional or state space model layers), we realize them as 1-D/2-D/N-D systems and exploit this structure as well. We distinguish ourselves from related work on Lipschitz constant estimation by more extensive structure exploitation (scalability) and a generalization to a large class of common neural network architectures. To show the versatility and computational advantages of our method, we apply it to different neural network architectures trained on MNIST and CIFAR-10.

Recently, semidefinite programming (SDP) techniques have shown great promise in providing accurate Lipschitz bounds for neural networks. Specifically, the LipSDP approach (Fazlyab et al., 2019) has received much attention and provides the least conservative Lipschitz upper bounds that can be computed with polynomial time guarantees. However, one main restriction of LipSDP is that its formulation requires the activation functions to be slope-restricted on $[0,1]$, preventing its further use for more general activation functions such as GroupSort, MaxMin, and Householder. One can rewrite MaxMin activations for example as residual ReLU networks. However, a direct application of LipSDP to the resultant residual ReLU networks is conservative and even fails in recovering the well-known fact that the MaxMin activation is 1-Lipschitz. Our paper bridges this gap and extends LipSDP beyond slope-restricted activation functions. To this end, we provide novel quadratic constraints for GroupSort, MaxMin, and Householder activations via leveraging their underlying properties such as sum preservation. Our proposed analysis is general and provides a unified approach for estimating $\ell_2$ and $\ell_\infty$ Lipschitz bounds for a rich class of neural network architectures, including non-residual and residual neural networks and implicit models, with GroupSort, MaxMin, and Householder activations. Finally, we illustrate the utility of our approach with a variety of experiments and show that our proposed SDPs generate less conservative Lipschitz bounds in comparison to existing approaches.

Lipschitz constants of neural networks allow for guarantees of robustness in image classification, safety in controller design, and generalizability beyond the training data. As calculating Lipschitz constants is NP-hard, techniques for estimating Lipschitz constants must navigate the trade-off between scalability and accuracy. In this work, we significantly push the scalability frontier of a semidefinite programming technique known as LipSDP while achieving zero accuracy loss. We first show that LipSDP has chordal sparsity, which allows us to derive a chordally sparse formulation that we call Chordal-LipSDP. The key benefit is that the main computational bottleneck of LipSDP, a large semidefinite constraint, is now decomposed into an equivalent collection of smaller ones: allowing Chordal-LipSDP to outperform LipSDP particularly as the network depth grows. Moreover, our formulation uses a tunable sparsity parameter that enables one to gain tighter estimates without incurring a significant computational cost. We illustrate the scalability of our approach through extensive numerical experiments.