Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant suffer from either lack of accuracy or poor scalability. In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently. Our main idea is to interpret activation functions as gradients of convex potential functions. Hence, they satisfy certain properties that can be described by quadratic constraints.

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.

Even though other optimization-based certification of Lipschitz constants have been proposed before, the theoretical result leading to the SDP formulation (Theorem 1) is novel. The multiple variants of the main algorithm aim to provide a more scalable method and in part succeeds at doing so (evaluated networks are still relatively simple), and despite the loss in accuracy the less-complex version of the methodology can achieve better or competitive estimation of the constant. On the other hand the authors overplay the fact that any method that estimates a Lipschitz constant on a multilayer network can be trivially parallelized by splitting the network into "chunks". This is not a particular advantage of their method and so I think the claims about the parallel version should be toned down. Even from the trivial upper bound on the Lipschitz constant, given by the product of the layer-wise constants, it is clear that such methods can be easily parallelized.

Congratulations on your work which the reviewers all appreciated. Please take the time to address their concerns for the final version of your paper.

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant suffer from either lack of accuracy or poor scalability. In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently. Our main idea is to interpret activation functions as gradients of convex potential functions. Hence, they satisfy certain properties that can be described by quadratic constraints.

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant suffer from either lack of accuracy or poor scalability. In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently. Our main idea is to interpret activation functions as gradients of convex potential functions. Hence, they satisfy certain properties that can be described by quadratic constraints.