On the Invariants of Softmax Attention

TL;DR

This paper uncovers invariant properties of softmax attention, revealing fundamental structural regularities across models and architectures with practical implications for analysis and training.

Contribution

It introduces the concept of the energy field and identifies invariant properties at both mechanism and model levels in softmax attention.

Findings

Energy field exhibits invariant properties across models and inputs.

Key incoherence leads to delocalization of attention variance.

Rank bounds confine the energy field to a low-dimensional subspace.

Abstract

Softmax attention maps every query--key interaction into a probability distribution, but the underlying structure remains largely unexplored. We define the \emph{energy field}, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs. Two classes of invariants emerge. \emph{Mechanism-level} invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. \emph{Model-level} regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call \emph{key incoherence}. These invariants have practical consequences. The rank bound confines the energy field to a low-dimensional subspace. Key incoherence yields a per-head training monitor. All results are verified at multiple context lengths and input texts.