Query lower bounds for log-concave sampling
This work addresses a fundamental gap in theoretical computer science by providing the first non-trivial lower bounds for log-concave sampling in higher dimensions, which is crucial for understanding the limits of algorithms in optimization and machine learning.
The paper tackles the problem of proving query lower bounds for log-concave sampling, establishing that sampling from strongly log-concave and log-smooth distributions in dimension d≥2 requires Ω(log κ) queries, which is sharp in constant dimensions, and sampling from Gaussians requires Ω̃(min(√κ log d, d)) queries, nearly sharp for Gaussians.
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 $Ω(\log κ)$ 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 $\widetilde Ω(\min(\sqrtκ\log d, d))$ queries, which is nearly sharp for the class of Gaussians. Here $κ$ 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.