Some Fundamental Aspects about Lipschitz Continuity of Neural Network Functions

Abstract

Lipschitz continuity is a simple yet crucial functional property of any predictive model for it lies at the core of the model's robustness, generalisation, as well as adversarial vulnerability. Our aim is to thoroughly investigate and characterise the Lipschitz behaviour of the functions realised by neural networks. Thus, we carry out an empirical investigation in a range of different settings (namely, architectures, losses, optimisers, label noise, and more) by exhausting the limits of the simplest and the most general lower and upper bounds. Although motivated primarily by computational hardness results, this choice nevertheless turns out to be rather resourceful and sheds light on several fundamental and intriguing traits of the Lipschitz continuity of neural network functions, which we also supplement with suitable theoretical arguments. As a highlight of this investigation, we identify a striking double descent trend in both upper and lower bounds to the Lipschitz constant with increasing network width -- which tightly aligns with the typical double descent trend in the test loss. Lastly, we touch upon the seeming (counter-intuitive) decline of the Lipschitz constant in the presence of label noise