Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension $d\ge 2$ requires $\Omega(\log \kappa)$ queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension $d$ (hence also from general log-concave and log-smooth distributions in dimension $d$) requires $\widetilde \Omega(\min(\sqrt\kappa \log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $\kappa$ denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in harmonic analysis, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature.

翻译：对数凹采样问题近年来在算法层面取得了显著进展，但该任务的查询下界证明仍具挑战性，此前仅在维度为一的情形下已知下界结果。本研究建立了以下查询下界：（1）在维度 $d\ge 2$ 中，从强对数凹且对数光滑分布中采样需要 $\Omega(\log \kappa)$ 次查询，该下界在任意常数维度下均是最优的；（2）从维度 $d$ 的高斯分布（进而从维度 $d$ 的一般对数凹且对数光滑分布）中采样需要 $\widetilde \Omega(\min(\sqrt\kappa \log d, d))$ 次查询，该下界对于高斯分布类几乎是最优的。其中 $\kappa$ 表示目标分布的条件数。我们的证明基于：（1）受调和分析中Kakeya猜想相关工作启发而构造的多尺度方法；（2）一种新颖的归约方法，证明块Krylov算法对该问题具有最优性，并与矩阵-向量查询文献中基于Wishart矩阵的下界技术建立了联系。