Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime

2026-05-11

AI Summary

This study investigates the concentration phenomenon of token distributions in deep encoder-only Transformers during inference, with a particular focus on their dynamical behavior in the low-temperature limit. By formulating a mean-field continuity equation and leveraging convergence analysis of multi-particle systems, the work provides the first quantitative characterization of the concentration rate and timescale of token distributions at low temperatures. Theoretically, using stability estimates in Wasserstein space, Lyapunov-type inequalities, and a quantitative Laplace principle, the authors prove that the Wasserstein distance of the token distribution converges at a rate of √(log(β+1)/β)·exp(Ct) + exp(−ct). Numerical experiments confirm rapid concentration within a log β timescale and reveal a terminal phase governed by the spectrum of the value matrix, elucidating the intrinsic connections between concentration dynamics, initial distributions, and attention-induced mappings.

Abstract

Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like $\sqrt{{\log(β+1)}/β}\exp(Ct)+\exp(-ct)$ in terms of the temperature parameter $β^{-1}\to 0$ and inference time $t\geq 0$. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as $t\to\infty$, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order $\logβ$ the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite $β$ and large $t$ the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.

Problem

• Transformers
• concentration phenomena
• mean-field limit
• low-temperature regime
• Wasserstein distance

Innovation

• mean-field transformers
• concentration phenomenon
• low-temperature regime
• Wasserstein stability
• self-attention dynamics