Parameterized Complexity of $L_p$-Lipschitz Constants for Input Convex Neural Networks and $L_p$-Norm Maximization over Zonotopes

Imagine you're developing a neural network that needs to be robust against small changes in input, like noise or slight variations in data. A key measure of this robustness is the Lipschitz constant, which quantifies how sensitive the network's output is to these perturbations. However, calculating this constant can be quite challenging, especially for more complex architectures. Current methods work well for certain types of networks, but they struggle with others, leading to what's known as computational hardness — a situation where finding a solution is extremely difficult or time-consuming. This is what's called W[1]-hardness, and it means that as the size of the problem grows, the time required to solve it can increase dramatically, making practical applications tough.

In this paper, the authors focus on a specific type of neural network called two-layer input-convex neural networks (ICNNs), which have properties that make them easier to analyze. They prove that for a range of norms, specifically the $L_p$ norms, maximizing the Lipschitz constant is computationally hard. By establishing this complexity, they provide a clearer understanding of the limitations and challenges faced when working with these networks. This work not only resolves an open question in the field but also emphasizes the importance of clear mathematical exposition, which can help builders like you navigate the complexities of neural network design more effectively.

The paper resolves an open problem in the complexity of Lipschitz constants for neural networks.

The results are supported by rigorous proofs and address previously unresolved questions.