Scaling Limits of Long-Context Transformers
For theorists studying transformer scaling, this provides precise phase transition boundaries and limiting laws, but the analysis is restricted to i.i.d. keys and fixed queries, limiting direct applicability.
This paper identifies the critical inverse temperature scale at which softmax attention transitions from uniform averaging to single-key collapse in long-context transformers, showing it scales as n^{2/(d-1)} for uniform keys on the sphere, and characterizes the limiting behavior of attention weights and output across all regimes.
We study the long-context limit of softmax self-attention with a fixed query and a random context of $n$ i.i.d. keys on the sphere, viewing the inverse temperature $β_n$ as the scaling parameter that decides whether attention degenerates into uniform averaging or collapses onto the single closest key. We show that the critical scale at which selectivity emerges is determined by the local exponent of the distance-to-query distribution near zero rather than by global features of the context, and scales like $β_n^\ast \asymp n^{2/(d-1)}$ for uniform keys on $\mathbb{S}^{d-1}$. Furthermore, we characterize the limiting laws of the ordered attention weights and of the attention output across all regimes of $β_n$: a subcritical regime in which the output reduces to a local average around $q$ with explicit deterministic bias and Gaussian fluctuations; a critical regime in which a finite collection of nearest keys retains macroscopic mass without single-key collapse; and a supercritical regime in which all mass concentrates on the closest key. Of notable interest is the subcritical case with identity value matrix where the attention map approximately implements a backward heat equation.