Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization

TL;DR

This paper provides a rigorous mathematical analysis of self-attention mechanisms in transformer architectures, focusing on mean-field models, gradient flows, and stationary points, revealing insights into their clustering and distribution behaviors.

Contribution

It introduces a novel gradient flow framework on the sphere for analyzing self-attention dynamics and explores the properties of stationary points and energy landscapes in this context.

Findings

Partial characterization of self-attention dynamics as gradient flows

Identification of stationary points related to energy minimizers and maximizers

Insights into clustering and uniform distribution patterns in transformer models

Abstract

The aim of this paper is to provide a mathematical analysis of transformer architectures using a self-attention mechanism with layer normalization. In particular, observed patterns in such architectures resembling either clusters or uniform distributions pose a number of challenging mathematical questions. We focus on a special case that admits a gradient flow formulation in the spaces of probability measures on the unit sphere under a special metric, which allows us to give at least partial answers in a rigorous way. The arising mathematical problems resemble those recently studied in aggregation equations, but with additional challenges emerging from restricting the dynamics to the sphere and the particular form of the interaction energy. We provide a rigorous framework for studying the gradient flow, which also suggests a possible metric geometry to study the general case (i.e. one that is not described by a gradient flow). We further analyze the stationary points of the induced self-attention dynamics. The latter are related to stationary points of the interaction energy in the Wasserstein geometry, and we further discuss energy minimizers and maximizers in different parameter settings.