Scaling Limits of Long-Context Transformers
Giuseppe Bruno, Shi Chen, Zhengjiang Lin, Yury Polyanskiy, Philippe Rigollet
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 $β_n$ 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.
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