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 geometric measure theory, 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：近年来，对数凹抽样在算法上取得了显著进展，但证明该任务下界的相应问题仍难以解决，此前仅在单一维度上已知下界。本文建立了以下查询下界：（1）在维度$d\ge 2$上从强对数凹和对数光滑分布中采样需要$\Omega(\log \kappa)$次查询，这一结果在任何常数维度中都是紧的；（2）在维度$d$上从高斯分布（因此也适用于维度$d$上的一般对数凹和对数光滑分布）中采样需要$\widetilde \Omega(\min(\sqrt\kappa \log d, d))$次查询，这一结果对于高斯分布类几乎是紧的。其中$\kappa$表示目标分布的 condition number（条件数）。我们的证明依赖于：（1）受几何测度论中Kakeya猜想研究启发的多尺度构造，以及（2）一种新颖的归约方法，该方法表明块Krylov算法对该问题是最优的，同时与矩阵-向量查询文献中基于Wishart矩阵开发的下界技术相关联。