On the Regularity of Attention
This work addresses a foundational mathematical understanding of attention mechanisms, which are widely used in AI, but it is incremental as it builds on existing theory to provide new proofs.
The paper tackled the problem of quantifying the smoothness (regularity) of the attention operation in neural networks, and proved that attention is Lipschitz continuous on compact domains with an estimated constant, extending results to non-compact domains for specific types.
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.