Query lower bounds for log-concave sampling
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Author(s)
Chewi, Sinho
de Dios Pont, Jaume
Li, Jerry
Lu, Chen
Narayanan, Shyam
Date Issued
June 21, 2024
Journal
Journal of the ACM
Publisher
Association for Computing Machinery
Citation
Chewi, Sinho, de Dios Pont, Jaume, Li, Jerry, Lu, Chen and Narayanan, Shyam. 2024. "Query lower bounds for log-concave sampling." Journal of the ACM.
Version
Final published version
Abstract
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 ≥ 2 requires Ω(log) queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in
dimension (hence also from general log-concave and log-smooth distributions in dimension) requires Ωe(min(
√ log,))
queries, which is nearly sharp for the class of Gaussians. Here 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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3673651
