Query-Oblivious Coresets for Softmax Attention: Improved Bounds and Efficient Constructions

-cross Abstract: A query-oblivious coreset for a softmax-attention head is a subset of the key-value pairs whose attention output is within $\varepsilon$ of the full one for every query in a ball. Liberty, Andoni and Kleiner proved that unweighted coresets of size $O(\sqrt d e^{\rho+\frac12\log\rho+o(\log\log\rho)}/\varepsilon)$ exist, $\rho$ the query radius times the centred key radius, against a lower bound $\Omega(\sqrt d e^{\rho}/\varepsilon)$, and conjectured that closing the gap needs new techniques. It does not: a spherical lift of both balls into one exponential-kernel instance lets the Bozzai-Rothvoss chaining bound apply, and Chevet's inequality splits key from value dimension, giving coresets of size $O(e^{\rho}(\sqrt{d_v}+\sqrt{d_k\log(1+\rho)})/\varepsilon)$ in randomised polynomial time, the first constructive whole-ball guarantee within $\sqrt{\log(1+\rho)}$ of the lower bound. A sampling cap $O(e^{2\rho}/\varepsilon^{2})$ completes the envelope; in fixed dimension Tai's diameter-free bound removes the logarithm, settling the Gaussian-restriction case of a Bozzai-Rothvoss question for the kernels. We give theLiberty-Andoni-Kleiner lower boud transfer the one-waycommunication bounds of Chen et r is the price of one signing forall queries. A census of every head of Qwen2.5-7B-Instruct and Llama-3-8B-Instruct finds $\rho$ at least 23.877, so everyactor $e^{\rho}/\varepsilon$prescribes a coreset larger than the cache: the algorithmic contribution is asymptotic on these models.