  Attention is a powerful component of modern neural networks across a wide
variety of domains. In this paper, we seek to quantify the regularity (i.e. the
amount of smoothness) of the attention operation. To accomplish this goal, we
propose a new mathematical framework that uses measure theory and integral
operators to model attention. We show that this framework is consistent with
the usual definition, and that it captures the essential properties of
attention. Then we use this framework to prove that, on compact domains, the
attention operation is Lipschitz continuous and provide an estimate of its
Lipschitz constant. Additionally, by focusing on a specific type of attention,
we extend these Lipschitz continuity results to non-compact domains. We also
discuss the effects regularity can have on NLP models, and applications to
invertible and infinitely-deep networks.
