  We propose a probabilistic interpretation of exponential dot product
attention of transformers and contrastive learning based off of exponential
families. The attention sublayer of transformers is equivalent to a gradient
ascent step of the log normalizer, which is the log-sum-exp term in the
Hopfield theory of attention. This ascent step induces a parallel expansion of
points, which is counterbalanced by a contraction from layer normalization. We
also state theoretical limitations of our theory and the Hopfield theory and
suggest directions for resolution.
