analiza aiqbench
This is a theoretical analysis of softmax attention in the long-context limit, using a fixed query and n random keys on a sphere. It identifies a critical inverse-temperature scale, roughly n^(2/(d-1)) for uniform keys, where attention shifts from averaging to selecting a few nearest keys, and it characterizes the limiting behavior in each regime. The results are asymptotic and rest on idealized assumptions (i.i.d. keys, fixed intrinsic dimension, uniform keys on the sphere), so they are not yet a direct guide to trained models with real token distributions. Readers interested in attention temperature scaling or long-context behavior should find the critical-scale result the most transferable part.
Scaling Limits of Long-Context Transformers
arXiv:2605.08505v2 Announce Type: replace Abstract: 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 with the intrinsic dimension of the key distribution held fixed, viewing the inverse temperature $\beta_n$ 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 $\beta_n^\ast \asymp n^{2/(d-1)}$ 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 $\beta_n$: 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.
Abstract: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 with the intrinsic dimension of the key distribution held fixed, viewing the inverse temperature $\beta_n$ 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 $\beta_n^\ast \asymp n^{2/(d-1)}$ 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 $\beta_n$: 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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Probability (math.PR); Statistics Theory (math.ST)
arXiv:2605.08505 [cs.LG]
(or arXiv:2605.08505v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.08505 Focus to learn more arXiv-issued DOI via DataCite
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